Theorem br1steq

Description: Uniqueness condition for the binary relation 1st . (Contributed by Scott Fenton, 11-Apr-2014) (Proof shortened by Mario Carneiro, 3-May-2015)

	Ref	Expression
Hypotheses	br1steq.1	$$⊢A\in \mathrm{V}$$
	br1steq.2	$$⊢B\in \mathrm{V}$$
Assertion	br1steq	$$⊢⟨A,B⟩1^{st}C\leftrightarrow C=A$$

Proof

Step	Hyp	Ref	Expression
1		br1steq.1	$$⊢A\in \mathrm{V}$$
2		br1steq.2	$$⊢B\in \mathrm{V}$$
3		br1steqg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(⟨A,B⟩1^{st}C\leftrightarrow C=A\right)$$
4	1 2 3	mp2an	$$⊢⟨A,B⟩1^{st}C\leftrightarrow C=A$$