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Theorem altopelaltxp 33432
Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5586, 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 33431 . 2 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫)
2 reeanv 3368 . . 3 (∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
3 eqcom 2828 . . . . 5 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫)
4 vex 3498 . . . . . 6 𝑥 ∈ V
5 vex 3498 . . . . . 6 𝑦 ∈ V
64, 5altopth 33425 . . . . 5 (⟪𝑥, 𝑦⟫ = ⟪𝑋, 𝑌⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
73, 6bitri 277 . . . 4 (⟪𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
872rexbii 3248 . . 3 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑋𝑦 = 𝑌))
9 risset 3267 . . . 4 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
10 risset 3267 . . . 4 (𝑌𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑌)
119, 10anbi12i 628 . . 3 ((𝑋𝐴𝑌𝐵) ↔ (∃𝑥𝐴 𝑥 = 𝑋 ∧ ∃𝑦𝐵 𝑦 = 𝑌))
122, 8, 113bitr4i 305 . 2 (∃𝑥𝐴𝑦𝐵𝑋, 𝑌⟫ = ⟪𝑥, 𝑦⟫ ↔ (𝑋𝐴𝑌𝐵))
131, 12bitri 277 1 (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1533  wcel 2110  wrex 3139  caltop 33412   ×× caltxp 33413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4562  df-pr 4564  df-altop 33414  df-altxp 33415
This theorem is referenced by: (None)
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