Theorem altopthb 35475

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 35474 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Hypotheses

Ref	Expression
altopthb.1	⊢ 𝐴 ∈ V
altopthb.2	⊢ 𝐷 ∈ V

Assertion

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

Proof of Theorem altopthb

Step	Hyp	Ref	Expression
1		altopthb.1	. 2 ⊢ 𝐴 ∈ V
2		altopthb.2	. 2 ⊢ 𝐷 ∈ V
3		altopthbg 35473	. 2 ⊢ ((𝐴 ∈ 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⟪caltop 35461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-altop 35463

This theorem is referenced by:  altopthc  35476