Equilibrium
System in equilibrium
It is a fact of experience that when two systems come into thermal contact (energy is transferred with a mechanism different from work, Heat) generally changes happen to both of them.
However if we isolate them we will see that eventually they reach a state in which no further change is perceptible, no matter how long one waits.
For simple systems meaning that they have no magnetic or electrical properties their equilibrium state can be completely described by their volume V and pressure P.
Systems in equilibrium with each other
If we have two systems $S_1$ and $S_2$ in thermal contact and we consider the two as one system $S_{12}$, then if the system $S_{12}$ is in equilibrium then they two systems $S_1$ and $S_2$ are said to be in equilibrium with each other.
Zeroth Law
The zeroth law states that if two systems A and C are in thermal equilibrium with a third one B(which we later will see that can be defined as the thermometer) then they are in equilibrium with one another as well.
Criterion of Equilibrium and Empirical Temperature
Suppose we have three systems A,B and C that are in equilibrium states.
$$A(P_A,V_A),B(P_B,V_B),C(P_C,V_C)$$
The condition under which A and B are in equilibrium may be expressed bu the equation:
$$F_1(P_A,V_A,P_B,V_B)=0 \Rightarrow P_B = f_1(P_A,V_A,V_B) $$
The same applies for the equilibrium of systems C and B.
$$P_B = f_2(P_C,V_C,V_B) $$
From the two above equations turns out that:
$$f_1(P_A,V_A,V_B)=f_2(P_C,V_C,V_B) \Rightarrow G(P_A,V_A,V_B,P_C,V_C)=0 $$
From the zeroth law turns out that if A and C are separately in equilibrium with B the they are with each other as well. Therefore:
$$ F_3(V_A,P_A,P_C,V_C) $$
Since the last two equations are to be equivalent, then the implicit function G is independent from $V_B$. Therefore going backwards:
$$ φ_1(P_A,V_A)=φ_2(P_C,V_C)$$
Therefore we can define a function
$$ φ(P,V)=θ$$
Which we call empirical temperature and it has the property that systems that take the same value are in equilibrium with each other.
Your case
We have defined that water boils at $100^oC$. Therefore according to the property of the emperical temperature function every system at $100^oC$ is in equilibrium with boiling water.