Theorem opthne 5466

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 5465	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
4	1, 2, 3	mp2an 705	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  ⟨cop 4597

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598

This theorem is used by:  xpord2lem  8140  xpord2pred  8143  xpord2indlem  8145  m2detleib  22818  addsqnreup  27638  mulsval  28333  gpgedg2ov  48864  gpgedg2iv  48865  gpg5nbgrvtx03starlem1  48866  gpg5nbgrvtx03starlem2  48867  gpg5nbgrvtx03starlem3  48868  gpg5nbgrvtx13starlem1  48869  gpg5nbgrvtx13starlem2  48870  gpg5nbgrvtx13starlem3  48871  gpg3nbgrvtx0  48874  gpg3nbgrvtx0ALT  48875  gpg3nbgrvtx1  48876  gpg3kgrtriex  48887  gpgprismgr4cycllem2  48894  gpgprismgr4cycllem7  48899  gpg5edgnedg  48928  zlmodzxzldeplem  49311  line2x  49567  inlinecirc02plem  49599