Theorem opth1 4394

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

Hypotheses

Ref	Expression
opth1.1	|- A e. _V
opth1.2	|- B e. _V

Assertion

Ref	Expression
opth1	|- ( <. A , B >. = <. C , D >. -> A = C )

Proof of Theorem opth1

Step	Hyp	Ref	Expression
1		opth1.1	. . . 4 |- A e. _V
2	1	sneqr 3926	. . 3 |- ( { A } = { C } -> A = C )
3	2	a1i 11	. 2 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C } -> A = C ) )
4		opth1.2	. . . . . . . . 9 |- B e. _V
5	1, 4	opi1 4390	. . . . . . . 8 |- { A } e. <. A , B >.
6		id 20	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
7	5, 6	syl5eleq 2490	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
8		oprcl 3968	. . . . . . 7 |- ( { A } e. <. C , D >. -> ( C e. _V /\ D e. _V ) )
9	7, 8	syl 16	. . . . . 6 |- ( <. A , B >. = <. C , D >. -> ( C e. _V /\ D e. _V ) )
10	9	simpld 446	. . . . 5 |- ( <. A , B >. = <. C , D >. -> C e. _V )
11		prid1g 3870	. . . . 5 |- ( C e. _V -> C e. { C , D } )
12	10, 11	syl 16	. . . 4 |- ( <. A , B >. = <. C , D >. -> C e. { C , D } )
13		eleq2 2465	. . . 4 |- ( { A } = { C , D } -> ( C e. { A } <-> C e. { C , D } ) )
14	12, 13	syl5ibrcom 214	. . 3 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C , D } -> C e. { A } ) )
15		elsni 3798	. . . 4 |- ( C e. { A } -> C = A )
16	15	eqcomd 2409	. . 3 |- ( C e. { A } -> A = C )
17	14, 16	syl6 31	. 2 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C , D } -> A = C ) )
18		dfopg 3942	. . . . 5 |- ( ( C e. _V /\ D e. _V ) -> <. C , D >. = { { C } , { C , D } } )
19	7, 8, 18	3syl 19	. . . 4 |- ( <. A , B >. = <. C , D >. -> <. C , D >. = { { C } , { C , D } } )
20	7, 19	eleqtrd 2480	. . 3 |- ( <. A , B >. = <. C , D >. -> { A } e. { { C } , { C , D } } )
21		elpri 3794	. . 3 |- ( { A } e. { { C } , { C , D } } -> ( { A } = { C } \/ { A } = { C , D } ) )
22	20, 21	syl 16	. 2 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C } \/ { A } = { C , D } ) )
23	3, 17, 22	mpjaod 371	1 |- ( <. A , B >. = <. C , D >. -> A = C )

Syntax hints:   -> wi 4   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   {csn 3774   {cpr 3775   <.cop 3777

This theorem is referenced by:  opth  4395  dmsnopg  5300  funcnvsn  5455  oprabid  6064  seqomlem2  6667  unxpdomlem3  7274  dfac5lem4  7963  dcomex  8283  canthwelem  8481  uzrdgfni  11253  gsum2d2  15503  2trllemA  21503  2pthon  21555  2pthon3v  21557  constr3lem2  21586

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783