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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > altxpexg | Structured version Visualization version GIF version |
Description: The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.) |
Ref | Expression |
---|---|
altxpexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ×× 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | altxpsspw 34552 | . 2 ⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) | |
2 | pwexg 5333 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ V) | |
3 | unexg 7682 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V) | |
4 | 2, 3 | sylan2 593 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V) |
5 | pwexg 5333 | . . 3 ⊢ ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) | |
6 | pwexg 5333 | . . 3 ⊢ (𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) | |
7 | 4, 5, 6 | 3syl 18 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) |
8 | ssexg 5280 | . 2 ⊢ (((𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) → (𝐴 ×× 𝐵) ∈ V) | |
9 | 1, 7, 8 | sylancr 587 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ×× 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3445 ∪ cun 3908 ⊆ wss 3910 𝒫 cpw 4560 ×× caltxp 34532 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
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 2714 df-cleq 2728 df-clel 2814 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-pw 4562 df-sn 4587 df-pr 4589 df-uni 4866 df-altop 34533 df-altxp 34534 |
This theorem is referenced by: (None) |
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