Theorem opthne 5462

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 ∈ V
opthne.2	⊢ 𝐵 ∈ 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 ∈ V
2		opthne.2	. 2 ⊢ 𝐵 ∈ V
3		opthneg 5461	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
4	1, 2, 3	mp2an 704	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))

Syntax hints:   ↔ wb 209   ∨ wo 860   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ⟨cop 4597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598

This theorem is referenced by:  xpord2lem  8134  xpord2pred  8137  xpord2indlem  8139  m2detleib  22753  addsqnreup  27569  mulsval  28264  gpgedg2ov  48715  gpgedg2iv  48716  gpg5nbgrvtx03starlem1  48717  gpg5nbgrvtx03starlem2  48718  gpg5nbgrvtx03starlem3  48719  gpg5nbgrvtx13starlem1  48720  gpg5nbgrvtx13starlem2  48721  gpg5nbgrvtx13starlem3  48722  gpg3nbgrvtx0  48725  gpg3nbgrvtx0ALT  48726  gpg3nbgrvtx1  48727  gpg3kgrtriex  48738  gpgprismgr4cycllem2  48745  gpgprismgr4cycllem7  48750  gpg5edgnedg  48779  zlmodzxzldeplem  49158  line2x  49414  inlinecirc02plem  49446