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

Theorem altopelaltxp 35977
Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5721, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopelaltxp (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))

Proof of Theorem altopelaltxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elaltxp 35976 . 2 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫)
2 reeanv 3229 . . 3 (∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
3 eqcom 2744 . . . . 5 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫)
4 vex 3484 . . . . . 6 𝑥 ∈ V
5 vex 3484 . . . . . 6 𝑦 ∈ V
64, 5altopth 35970 . . . . 5 (⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
73, 6bitri 275 . . . 4 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
872rexbii 3129 . . 3 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌))
9 risset 3233 . . . 4 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
10 risset 3233 . . . 4 (𝑌𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑌)
119, 10anbi12i 628 . . 3 ((𝑋𝐴𝑌𝐵) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
122, 8, 113bitr4i 303 . 2 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑋𝐴𝑌𝐵))
131, 12bitri 275 1 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  caltop 35957   ×× caltxp 35958
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-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-altop 35959  df-altxp 35960
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator