Theorem opthne 4951

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

Hypotheses

Ref	Expression
opthne.1	|- A e. _V
opthne.2	|- B e. _V

Assertion

Ref	Expression
opthne	|- ( <. A , B >. =/= <. C , D >. <-> ( A =/= C \/ B =/= D ) )

Proof of Theorem opthne

Step	Hyp	Ref	Expression
1		opthne.1	. 2 |- A e. _V
2		opthne.2	. 2 |- B e. _V
3		opthneg 4950	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. =/= <. C , D >. <-> ( A =/= C \/ B =/= D ) ) )
4	1, 2, 3	mp2an 708	1 |- ( <. A , B >. =/= <. C , D >. <-> ( A =/= C \/ B =/= D ) )

Syntax hints:   <-> wb 196   \/ wo 383   e. wcel 1990   =/= wne 2794   _Vcvv 3200   <.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:  m2detleib  20437  zlmodzxzldeplem  42287