Theorem opth2 5395

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 5394	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ 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   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568

This theorem is referenced by:  eqvinop  5401  opelxp  5625  fsn  7007  opiota  7899  canthwe  10407  ltresr  10896  mat1dimelbas  21620  fmucndlem  23443  hgt750lemb  32636  diblsmopel  39185  cdlemn7  39217  dihordlem7  39228  xihopellsmN  39268  dihopellsm  39269  dihpN  39350