Theorem opth2 5428

Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)

Hypotheses

Ref	Expression
opth2.1	⊢ 𝐶 ∈ V
opth2.2	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
opth2	⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opth2

Step	Hyp	Ref	Expression
1		opth2.1	. 2 ⊢ 𝐶 ∈ V
2		opth2.2	. 2 ⊢ 𝐷 ∈ V
3		opthg2 5427	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ 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   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587

This theorem is referenced by:  eqvinop  5435  opelxp  5660  fsn  7080  opiota  8003  canthwe  10562  ltresr  11051  mat1dimelbas  22415  fmucndlem  24234  hgt750lemb  34813  diblsmopel  41431  cdlemn7  41463  dihordlem7  41474  xihopellsmN  41514  dihopellsm  41515  dihpN  41596  cofidvala  49361  cofidval  49364