Theorem opthne 5473

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 5472	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
4	1, 2, 3	mp2an 689	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))

Syntax hints:   ↔ wb 205   ∨ wo 844   ∈ wcel 2098   ≠ wne 2932  Vcvv 3466  ⟨cop 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628

This theorem is referenced by:  xpord2lem  8123  xpord2pred  8126  xpord2indlem  8128  m2detleib  22477  addsqnreup  27317  mulsval  27950  zlmodzxzldeplem  47428  line2x  47689  inlinecirc02plem  47721