Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 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)) | |
4 | 2, 3 | mpbiri 257 | . . . 4 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V) |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)) |
6 | 5 | rexlimivv 3194 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V) |
7 | eqeq1 2741 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫)) | |
8 | 7 | 2rexbidv 3211 | . . 3 ⊢ (𝑧 = 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)) |
9 | df-altxp 34482 | . . 3 ⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫} | |
10 | 8, 9 | elab2g 3630 | . 2 ⊢ (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)) |
11 | 1, 6, 10 | pm5.21nii 379 | 1 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 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 |
Copyright terms: Public domain | W3C validator |