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Theorem elaltxp 32621
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 3429 . 2 (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2 altopex 32606 . . . . 5 𝑥, 𝑦⟫ ∈ V
3 eleq1 2894 . . . . 5 (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
42, 3mpbiri 250 . . . 4 (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
54a1i 11 . . 3 ((𝑥𝐴𝑦𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
65rexlimivv 3246 . 2 (∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7 eqeq1 2829 . . . 4 (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
872rexbidv 3267 . . 3 (𝑧 = 𝑋 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9 df-altxp 32605 . . 3 (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
108, 9elab2g 3574 . 2 (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
111, 6, 10pm5.21nii 370 1 (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wrex 3118  Vcvv 3414  caltop 32602   ×× caltxp 32603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-dif 3801  df-un 3803  df-nul 4145  df-sn 4398  df-pr 4400  df-altop 32604  df-altxp 32605
This theorem is referenced by:  altopelaltxp  32622  altxpsspw  32623
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