Theorem altopth 36142

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 5423), 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 36140	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4	1, 2, 3	mp2an 693	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3439  ⟪caltop 36129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-sn 4580  df-pr 4582  df-altop 36131

This theorem is referenced by:  altopthd  36145  altopelaltxp  36149