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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 5156 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | 1 | unissi 4397 | . . . 4 ⊢ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) |
3 | unipw 4845 | . . . 4 ⊢ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) = 𝒫 (𝐴 ∪ 𝐵) | |
4 | 2, 3 | sseqtri 3600 | . . 3 ⊢ ∪ (𝐴 × 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
5 | 4 | unissi 4397 | . 2 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 (𝐴 ∪ 𝐵) |
6 | unipw 4845 | . 2 ⊢ ∪ 𝒫 (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵) | |
7 | 5, 6 | sseqtri 3600 | 1 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: relfld 5578 filnetlem3 31545 |
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