Theorem opth1 4710

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 4183	. . 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 4704	. . . . . . . 8 |- { A } e. <. A , B >.
6		id 22	. . . . . . . 8 |- ( <. A , B >. = <. C , D >. -> <. A , B >. = <. C , D >. )
7	5, 6	syl5eleq 2548	. . . . . . 7 |- ( <. A , B >. = <. C , D >. -> { A } e. <. C , D >. )
8		oprcl 4228	. . . . . . 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 457	. . . . 5 |- ( <. A , B >. = <. C , D >. -> C e. _V )
11		prid1g 4122	. . . . 5 |- ( C e. _V -> C e. { C , D } )
12	10, 11	syl 16	. . . 4 |- ( <. A , B >. = <. C , D >. -> C e. { C , D } )
13		eleq2 2527	. . . 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 4041	. . . 4 |- ( C e. { A } -> C = A )
16	15	eqcomd 2462	. . 3 |- ( C e. { A } -> A = C )
17	14, 16	syl6 33	. 2 |- ( <. A , B >. = <. C , D >. -> ( { A } = { C , D } -> A = C ) )
18		dfopg 4201	. . . . 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 2544	. . 3 |- ( <. A , B >. = <. C , D >. -> { A } e. { { C } , { C , D } } )
21		elpri 4036	. . 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 379	1 |- ( <. A , B >. = <. C , D >. -> A = C )

Syntax hints:   -> wi 4   \/ wo 366   /\ wa 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   {csn 4016   {cpr 4018   <.cop 4022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023

This theorem is referenced by:  opth  4711  dmsnopg  5462  funcnvsn  5615  oprabid  6297  seqomlem2  7108  unxpdomlem3  7719  dfac5lem4  8498  dcomex  8818  canthwelem  9017  uzrdgfni  12051  gsum2d2  17198  2trllemA  24754  2pthon  24806  2pthon3v  24808  constr3lem2  24848