Theorem opth1 4560

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 4035	. . 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 4554	. . . . . . . 8 |- { A } e. <. A , B >.
6		id 22	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
7	5, 6	syl5eleq 2524	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
8		oprcl 4079	. . . . . . 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 459	. . . . 5 |- ( <. A , B >. = <. C , D >. -> C e. _V )
11		prid1g 3976	. . . . 5 |- ( C e. _V -> C e. { C , D } )
12	10, 11	syl 16	. . . 4 |- ( <. A , B >. = <. C , D >. -> C e. { C , D } )
13		eleq2 2499	. . . 4 |- ( { A } = { C , D } -> ( C e. { A } <-> C e. { C , D } ) )
14	12, 13	syl5ibrcom 222	. . 3 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C , D } -> C e. { A } ) )
15		elsni 3897	. . . 4 |- ( C e. { A } -> C = A )
16	15	eqcomd 2443	. . 3 |- ( C e. { A } -> A = C )
17	14, 16	syl6 33	. 2 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C , D } -> A = C ) )
18		dfopg 4052	. . . . 5 |- ( ( C e. _V /\ D e. _V ) -> <. C , D >. = { { C } , { C , D } } )
19	7, 8, 18	3syl 20	. . . 4 |- ( <. A , B >. = <. C , D >. -> <. C , D >. = { { C } , { C , D } } )
20	7, 19	eleqtrd 2514	. . 3 |- ( <. A , B >. = <. C , D >. -> { A } e. { { C } , { C , D } } )
21		elpri 3892	. . 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 381	1 |- ( <. A , B >. = <. C , D >. -> A = C )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2967   {csn 3872   {cpr 3874   <.cop 3878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879

This theorem is referenced by:  opth  4561  dmsnopg  5305  funcnvsn  5458  oprabid  6110  seqomlem2  6898  unxpdomlem3  7511  dfac5lem4  8288  dcomex  8608  canthwelem  8809  uzrdgfni  11773  gsum2d2  16456  2trllemA  23417  2pthon  23469  2pthon3v  23471  constr3lem2  23500