Theorem opth1

Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Hypotheses	opth1.1	$$⊢A\in \mathrm{V}$$
	opth1.2	$$⊢B\in \mathrm{V}$$
Assertion	opth1	$$⊢⟨A,B⟩=⟨C,D⟩\to A=C$$

Proof

Step	Hyp	Ref	Expression
1		opth1.1	$$⊢A\in \mathrm{V}$$
2		opth1.2	$$⊢B\in \mathrm{V}$$
3	1 2	opi1	$$⊢\left\{A\right\}\in ⟨A,B⟩$$
4		id	$$⊢⟨A,B⟩=⟨C,D⟩\to ⟨A,B⟩=⟨C,D⟩$$
5	3 4	eleqtrid	$$⊢⟨A,B⟩=⟨C,D⟩\to \left\{A\right\}\in ⟨C,D⟩$$
6	1	sneqr	$$⊢\left\{A\right\}=\left\{C\right\}\to A=C$$
7	6	a1i	$$⊢\left\{A\right\}\in ⟨C,D⟩\to \left(\left\{A\right\}=\left\{C\right\}\to A=C\right)$$
8		oprcl	$$⊢\left\{A\right\}\in ⟨C,D⟩\to \left(C\in \mathrm{V}\wedge D\in \mathrm{V}\right)$$
9	8	simpld	$$⊢\left\{A\right\}\in ⟨C,D⟩\to C\in \mathrm{V}$$
10		prid1g	$$⊢C\in \mathrm{V}\to C\in \left\{C,D\right\}$$
11	9 10	syl	$$⊢\left\{A\right\}\in ⟨C,D⟩\to C\in \left\{C,D\right\}$$
12		eleq2	$$⊢\left\{A\right\}=\left\{C,D\right\}\to \left(C\in \left\{A\right\}\leftrightarrow C\in \left\{C,D\right\}\right)$$
13	11 12	syl5ibrcom	$$⊢\left\{A\right\}\in ⟨C,D⟩\to \left(\left\{A\right\}=\left\{C,D\right\}\to C\in \left\{A\right\}\right)$$
14		elsni	$$⊢C\in \left\{A\right\}\to C=A$$
15	14	eqcomd	$$⊢C\in \left\{A\right\}\to A=C$$
16	13 15	syl6	$$⊢\left\{A\right\}\in ⟨C,D⟩\to \left(\left\{A\right\}=\left\{C,D\right\}\to A=C\right)$$
17		id	$$⊢\left\{A\right\}\in ⟨C,D⟩\to \left\{A\right\}\in ⟨C,D⟩$$
18		dfopg	$$⊢\left(C\in \mathrm{V}\wedge D\in \mathrm{V}\right)\to ⟨C,D⟩=\left\{\left\{C\right\},\left\{C,D\right\}\right\}$$
19	8 18	syl	$$⊢\left\{A\right\}\in ⟨C,D⟩\to ⟨C,D⟩=\left\{\left\{C\right\},\left\{C,D\right\}\right\}$$
20	17 19	eleqtrd	$$⊢\left\{A\right\}\in ⟨C,D⟩\to \left\{A\right\}\in \left\{\left\{C\right\},\left\{C,D\right\}\right\}$$
21		elpri	$$⊢\left\{A\right\}\in \left\{\left\{C\right\},\left\{C,D\right\}\right\}\to \left(\left\{A\right\}=\left\{C\right\}\vee \left\{A\right\}=\left\{C,D\right\}\right)$$
22	20 21	syl	$$⊢\left\{A\right\}\in ⟨C,D⟩\to \left(\left\{A\right\}=\left\{C\right\}\vee \left\{A\right\}=\left\{C,D\right\}\right)$$
23	7 16 22	mpjaod	$$⊢\left\{A\right\}\in ⟨C,D⟩\to A=C$$
24	5 23	syl	$$⊢⟨A,B⟩=⟨C,D⟩\to A=C$$