Theorem opthneg 4950

Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)

Assertion

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

Proof of Theorem opthneg

Step	Hyp	Ref	Expression
1		df-ne 2795	. 2 |- ( <. A , B >. =/= <. C , D >. <-> -. <. A , B >. = <. C , D >. )
2		opthg 4946	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )
3	2	notbid 308	. . 3 |- ( ( A e. V /\ B e. W ) -> ( -. <. A , B >. = <. C , D >. <-> -. ( A = C /\ B = D ) ) )
4		ianor 509	. . . 4 |- ( -. ( A = C /\ B = D ) <-> ( -. A = C \/ -. B = D ) )
5		df-ne 2795	. . . . 5 |- ( A =/= C <-> -. A = C )
6		df-ne 2795	. . . . 5 |- ( B =/= D <-> -. B = D )
7	5, 6	orbi12i 543	. . . 4 |- ( ( A =/= C \/ B =/= D ) <-> ( -. A = C \/ -. B = D ) )
8	4, 7	bitr4i 267	. . 3 |- ( -. ( A = C /\ B = D ) <-> ( A =/= C \/ B =/= D ) )
9	3, 8	syl6bb 276	. 2 |- ( ( A e. V /\ B e. W ) -> ( -. <. A , B >. = <. C , D >. <-> ( A =/= C \/ B =/= D ) ) )
10	1, 9	syl5bb 272	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. =/= <. C , D >. <-> ( A =/= C \/ B =/= D ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   <.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-ne 2795  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:  opthne  4951  zlmodzxznm  42286