Theorem opkth 4133

Description: Two Kuratowski ordered pairs are equal iff their components are equal. (Contributed by SF, 12-Jan-2015.)

Hypotheses

Ref	Expression
opkth.1	⊢ A ∈ V
opkth.2	⊢ B ∈ V
opkth.3	⊢ D ∈ V

Assertion

Ref	Expression
opkth	⊢ (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C ∧ B = D))

Proof of Theorem opkth

Step	Hyp	Ref	Expression
1		opkth.1	. 2 ⊢ A ∈ V
2		opkth.2	. 2 ⊢ B ∈ V
3		opkth.3	. 2 ⊢ D ∈ V
4		opkthg 4132	. 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ D ∈ V) → (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C ∧ B = D)))
5	1, 2, 3, 4	mp3an 1277	1 ⊢ (⟪A, B⟫ = ⟪C, D⟫ ↔ (A = C ∧ B = D))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟪copk 4058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by: (None)