Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  altopthb Structured version   Visualization version   GIF version

Theorem altopthb 35971
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 35970 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 = 𝐶𝐵 = 𝐷))

Proof of Theorem altopthb
StepHypRef Expression
1 altopthb.1 . 2 𝐴 ∈ V
2 altopthb.2 . 2 𝐷 ∈ V
3 altopthbg 35969 . 2 ((𝐴 ∈ V ∧ 𝐷 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
41, 2, 3mp2an 692 1 (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  caltop 35957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-altop 35959
This theorem is referenced by:  altopthc  35972
  Copyright terms: Public domain W3C validator