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Theorem altopelaltxp 32459
Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5313, 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 32458 . 2 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫)
2 reeanv 3254 . . 3 (∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
3 eqcom 2772 . . . . 5 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫)
4 vex 3353 . . . . . 6 𝑥 ∈ V
5 vex 3353 . . . . . 6 𝑦 ∈ V
64, 5altopth 32452 . . . . 5 (⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
73, 6bitri 266 . . . 4 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
872rexbii 3189 . . 3 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌))
9 risset 3209 . . . 4 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
10 risset 3209 . . . 4 (𝑌𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑌)
119, 10anbi12i 620 . . 3 ((𝑋𝐴𝑌𝐵) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
122, 8, 113bitr4i 294 . 2 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑋𝐴𝑌𝐵))
131, 12bitri 266 1 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wrex 3056  caltop 32439   ×× caltxp 32440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-sn 4335  df-pr 4337  df-altop 32441  df-altxp 32442
This theorem is referenced by: (None)
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