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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 5646 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | 1 | unissi 4809 | . . . 4 ⊢ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) |
3 | unipw 5308 | . . . 4 ⊢ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) = 𝒫 (𝐴 ∪ 𝐵) | |
4 | 2, 3 | sseqtri 3951 | . . 3 ⊢ ∪ (𝐴 × 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
5 | 4 | unissi 4809 | . 2 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 (𝐴 ∪ 𝐵) |
6 | unipw 5308 | . 2 ⊢ ∪ 𝒫 (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵) | |
7 | 5, 6 | sseqtri 3951 | 1 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3879 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 × cxp 5517 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: relfld 6094 filnetlem3 33841 |
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