Theorem altopth 35696

Description: The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that 𝐶 and 𝐷 are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 5478), requires 𝐷 to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)

Hypotheses

Ref	Expression
altopth.1	⊢ 𝐴 ∈ V
altopth.2	⊢ 𝐵 ∈ V

Assertion

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

Proof of Theorem altopth

Step	Hyp	Ref	Expression
1		altopth.1	. 2 ⊢ 𝐴 ∈ V
2		altopth.2	. 2 ⊢ 𝐵 ∈ V
3		altopthg 35694	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4	1, 2, 3	mp2an 690	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461  ⟪caltop 35683

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 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-sn 4631  df-pr 4633  df-altop 35685

This theorem is referenced by:  altopthd  35699  altopelaltxp  35703