Theorem opth1 4553

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 4028	. . 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 4547	. . . . . . . 8 |- { A } e. <. A , B >.
6		id 22	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
7	5, 6	syl5eleq 2519	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
8		oprcl 4072	. . . . . . 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 456	. . . . 5 |- ( <. A , B >. = <. C , D >. -> C e. _V )
11		prid1g 3969	. . . . 5 |- ( C e. _V -> C e. { C , D } )
12	10, 11	syl 16	. . . 4 |- ( <. A , B >. = <. C , D >. -> C e. { C , D } )
13		eleq2 2494	. . . 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 3890	. . . 4 |- ( C e. { A } -> C = A )
16	15	eqcomd 2438	. . 3 |- ( C e. { A } -> A = C )
17	14, 16	syl6 33	. 2 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C , D } -> A = C ) )
18		dfopg 4045	. . . . 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 2509	. . 3 |- ( <. A , B >. = <. C , D >. -> { A } e. { { C } , { C , D } } )
21		elpri 3885	. . 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 1362   e. wcel 1755   _Vcvv 2962   {csn 3865   {cpr 3867   <.cop 3871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872

This theorem is referenced by:  opth  4554  dmsnopg  5298  funcnvsn  5451  oprabid  6104  seqomlem2  6892  unxpdomlem3  7507  dfac5lem4  8284  dcomex  8604  canthwelem  8805  uzrdgfni  11765  gsum2d2  16440  2trllemA  23272  2pthon  23324  2pthon3v  23326  constr3lem2  23355