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Theorem elaltxp 35941
Description: Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
Assertion
Ref Expression
elaltxp (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦

Proof of Theorem elaltxp
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3509 . 2 (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2 altopex 35926 . . . . 5 𝑥, 𝑦⟫ ∈ V
3 eleq1 2832 . . . . 5 (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
42, 3mpbiri 258 . . . 4 (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
54a1i 11 . . 3 ((𝑥𝐴𝑦𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
65rexlimivv 3207 . 2 (∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7 eqeq1 2744 . . . 4 (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
872rexbidv 3228 . . 3 (𝑧 = 𝑋 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9 df-altxp 35925 . . 3 (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
108, 9elab2g 3696 . 2 (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
111, 6, 10pm5.21nii 378 1 (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  wrex 3076  Vcvv 3488  caltop 35922   ×× caltxp 35923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651  df-altop 35924  df-altxp 35925
This theorem is referenced by:  altopelaltxp  35942  altxpsspw  35943
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