Theorem opthne 5437

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 5436	. 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 2109   ≠ wne 2925  Vcvv 3444  ⟨cop 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592

This theorem is referenced by:  xpord2lem  8098  xpord2pred  8101  xpord2indlem  8103  m2detleib  22494  addsqnreup  27330  mulsval  27988  gpgedg2ov  48030  gpgedg2iv  48031  gpg5nbgrvtx03starlem1  48032  gpg5nbgrvtx03starlem2  48033  gpg5nbgrvtx03starlem3  48034  gpg5nbgrvtx13starlem1  48035  gpg5nbgrvtx13starlem2  48036  gpg5nbgrvtx13starlem3  48037  gpg3nbgrvtx0  48040  gpg3nbgrvtx0ALT  48041  gpg3nbgrvtx1  48042  gpg3kgrtriex  48053  gpgprismgr4cycllem2  48059  gpgprismgr4cycllem7  48064  zlmodzxzldeplem  48460  line2x  48716  inlinecirc02plem  48748