Theorem opth2 5455

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 5454	. 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 1540   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608

This theorem is referenced by:  eqvinop  5462  opelxp  5690  fsn  7125  opiota  8058  canthwe  10665  ltresr  11154  mat1dimelbas  22409  fmucndlem  24229  hgt750lemb  34688  diblsmopel  41190  cdlemn7  41222  dihordlem7  41233  xihopellsmN  41273  dihopellsm  41274  dihpN  41355