Theorem altopth 36427

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 5458), 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 36425	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4	1, 2, 3	mp2an 704	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  Vcvv 3453  ⟪caltop 36414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-ss 3921  df-sn 4589  df-pr 4591  df-altop 36416

This theorem is referenced by:  altopthd  36430  altopelaltxp  36434