Theorem opthne 5447

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 5446	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
4	1, 2, 3	mp2an 702	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))

Syntax hints:   ↔ wb 208   ∨ wo 858   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453  ⟨cop 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586

This theorem is referenced by:  xpord2lem  8116  xpord2pred  8119  xpord2indlem  8121  m2detleib  22679  addsqnreup  27495  mulsval  28190  gpgedg2ov  48649  gpgedg2iv  48650  gpg5nbgrvtx03starlem1  48651  gpg5nbgrvtx03starlem2  48652  gpg5nbgrvtx03starlem3  48653  gpg5nbgrvtx13starlem1  48654  gpg5nbgrvtx13starlem2  48655  gpg5nbgrvtx13starlem3  48656  gpg3nbgrvtx0  48659  gpg3nbgrvtx0ALT  48660  gpg3nbgrvtx1  48661  gpg3kgrtriex  48672  gpgprismgr4cycllem2  48679  gpgprismgr4cycllem7  48684  gpg5edgnedg  48713  zlmodzxzldeplem  49081  line2x  49337  inlinecirc02plem  49369