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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 3360 | . 2 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V) | |
2 | altopex 32405 | . . . . 5 ⊢ ⟪𝑥, 𝑦⟫ ∈ V | |
3 | eleq1 2836 | . . . . 5 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V)) | |
4 | 2, 3 | mpbiri 248 | . . . 4 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V) |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)) |
6 | 5 | rexlimivv 3182 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V) |
7 | eqeq1 2773 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫)) | |
8 | 7 | 2rexbidv 3203 | . . 3 ⊢ (𝑧 = 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)) |
9 | df-altxp 32404 | . . 3 ⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫} | |
10 | 8, 9 | elab2g 3501 | . 2 ⊢ (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)) |
11 | 1, 6, 10 | pm5.21nii 367 | 1 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1629 ∈ wcel 2143 ∃wrex 3060 Vcvv 3348 ⟪caltop 32401 ×× caltxp 32402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1868 ax-4 1883 ax-5 1989 ax-6 2055 ax-7 2091 ax-9 2152 ax-10 2172 ax-11 2188 ax-12 2201 ax-13 2406 ax-ext 2749 ax-sep 4911 ax-nul 4919 ax-pr 5033 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1632 df-ex 1851 df-nf 1856 df-sb 2048 df-clab 2756 df-cleq 2762 df-clel 2765 df-nfc 2900 df-ral 3064 df-rex 3065 df-v 3350 df-dif 3723 df-un 3725 df-nul 4061 df-sn 4314 df-pr 4316 df-altop 32403 df-altxp 32404 |
This theorem is referenced by: altopelaltxp 32421 altxpsspw 32422 |
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