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Mirrors > Home > MPE Home > Th. List > Mathboxes > elaltxp | Structured version Visualization version GIF version |
Description: Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.) |
Ref | Expression |
---|---|
elaltxp | ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫) |
Step | Hyp | Ref | 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)) | |
4 | 2, 3 | mpbiri 250 | . . . 4 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V) |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)) |
6 | 5 | rexlimivv 3246 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V) |
7 | eqeq1 2829 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫)) | |
8 | 7 | 2rexbidv 3267 | . . 3 ⊢ (𝑧 = 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)) |
9 | df-altxp 32605 | . . 3 ⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫} | |
10 | 8, 9 | elab2g 3574 | . 2 ⊢ (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)) |
11 | 1, 6, 10 | pm5.21nii 370 | 1 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 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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