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| Mirrors > Home > MPE Home > Th. List > unixpss | Structured version Visualization version GIF version | ||
| Description: The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.) |
| Ref | Expression |
|---|---|
| unixpss | ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xpsspw 5819 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
| 2 | 1 | unissi 4916 | . . . 4 ⊢ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) |
| 3 | unipw 5455 | . . . 4 ⊢ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) = 𝒫 (𝐴 ∪ 𝐵) | |
| 4 | 2, 3 | sseqtri 4032 | . . 3 ⊢ ∪ (𝐴 × 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
| 5 | 4 | unissi 4916 | . 2 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 (𝐴 ∪ 𝐵) |
| 6 | unipw 5455 | . 2 ⊢ ∪ 𝒫 (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵) | |
| 7 | 5, 6 | sseqtri 4032 | 1 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: relfld 6295 filnetlem3 36381 |
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