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Theorem elaltxp 32207
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 3243 . 2 (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2 altopex 32192 . . . . 5 𝑥, 𝑦⟫ ∈ V
3 eleq1 2718 . . . . 5 (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
42, 3mpbiri 248 . . . 4 (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
54a1i 11 . . 3 ((𝑥𝐴𝑦𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
65rexlimivv 3065 . 2 (∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7 eqeq1 2655 . . . 4 (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
872rexbidv 3086 . . 3 (𝑧 = 𝑋 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9 df-altxp 32191 . . 3 (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
108, 9elab2g 3385 . 2 (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
111, 6, 10pm5.21nii 367 1 (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  caltop 32188   ×× caltxp 32189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-altop 32190  df-altxp 32191
This theorem is referenced by:  altopelaltxp  32208  altxpsspw  32209
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