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Theorem elaltxp 34498
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 3461 . 2 (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2 altopex 34483 . . . . 5 𝑥, 𝑦⟫ ∈ V
3 eleq1 2825 . . . . 5 (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
42, 3mpbiri 257 . . . 4 (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
54a1i 11 . . 3 ((𝑥𝐴𝑦𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
65rexlimivv 3194 . 2 (∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7 eqeq1 2741 . . . 4 (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
872rexbidv 3211 . . 3 (𝑧 = 𝑋 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9 df-altxp 34482 . . 3 (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
108, 9elab2g 3630 . 2 (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
111, 6, 10pm5.21nii 379 1 (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3071  Vcvv 3443  caltop 34479   ×× caltxp 34480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-rex 3072  df-v 3445  df-dif 3911  df-un 3913  df-nul 4281  df-sn 4585  df-pr 4587  df-altop 34481  df-altxp 34482
This theorem is referenced by:  altopelaltxp  34499  altxpsspw  34500
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