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Theorem altopelaltxp 31746
Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5108, 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 31745 . 2 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫)
2 reeanv 3097 . . 3 (∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
3 eqcom 2628 . . . . 5 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫)
4 vex 3189 . . . . . 6 𝑥 ∈ V
5 vex 3189 . . . . . 6 𝑦 ∈ V
64, 5altopth 31739 . . . . 5 (⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
73, 6bitri 264 . . . 4 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
872rexbii 3035 . . 3 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌))
9 risset 3055 . . . 4 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
10 risset 3055 . . . 4 (𝑌𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑌)
119, 10anbi12i 732 . . 3 ((𝑋𝐴𝑌𝐵) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
122, 8, 113bitr4i 292 . 2 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑋𝐴𝑌𝐵))
131, 12bitri 264 1 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  caltop 31726   ×× caltxp 31727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-sn 4151  df-pr 4153  df-altop 31728  df-altxp 31729
This theorem is referenced by: (None)
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