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Theorem altopelaltxp 31253
 Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5070, 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 31252 . 2 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫)
2 reeanv 3086 . . 3 (∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
3 eqcom 2617 . . . . 5 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫)
4 vex 3176 . . . . . 6 𝑥 ∈ V
5 vex 3176 . . . . . 6 𝑦 ∈ V
64, 5altopth 31246 . . . . 5 (⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
73, 6bitri 263 . . . 4 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
872rexbii 3024 . . 3 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌))
9 risset 3044 . . . 4 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
10 risset 3044 . . . 4 (𝑌𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑌)
119, 10anbi12i 729 . . 3 ((𝑋𝐴𝑌𝐵) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
122, 8, 113bitr4i 291 . 2 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑋𝐴𝑌𝐵))
131, 12bitri 263 1 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ⟪caltop 31233   ×× caltxp 31234 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-altop 31235  df-altxp 31236 This theorem is referenced by: (None)
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