Theorem opthne 5486

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 5485	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
4	1, 2, 3	mp2an 692	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))

Syntax hints:   ↔ wb 206   ∨ wo 847   ∈ wcel 2107   ≠ wne 2939  Vcvv 3479  ⟨cop 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632

This theorem is referenced by:  xpord2lem  8168  xpord2pred  8171  xpord2indlem  8173  m2detleib  22638  addsqnreup  27488  mulsval  28136  gpg5nbgrvtx03starlem1  48029  gpg5nbgrvtx03starlem2  48030  gpg5nbgrvtx03starlem3  48031  gpg5nbgrvtx13starlem1  48032  gpg5nbgrvtx13starlem2  48033  gpg5nbgrvtx13starlem3  48034  gpg3nbgrvtx0  48037  gpg3nbgrvtx0ALT  48038  gpg3nbgrvtx1  48039  gpg3kgrtriex  48050  zlmodzxzldeplem  48420  line2x  48680  inlinecirc02plem  48712