Theorem opth1g 4947

Description: Equality of the first members of equal ordered pairs. Closed form of opth1 4944. (Contributed by AV, 14-Oct-2018.)

Assertion

Ref	Expression
opth1g	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. -> A = C ) )

Proof of Theorem opth1g

Step	Hyp	Ref	Expression
1		opthg 4946	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )
2		simpl 473	. 2 |- ( ( A = C /\ B = D ) -> A = C )
3	1, 2	syl6bi 243	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. -> A = C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  wrdlen2i  13686