![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5815 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | 1 | unissi 4921 | . . . 4 ⊢ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) |
3 | unipw 5456 | . . . 4 ⊢ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) = 𝒫 (𝐴 ∪ 𝐵) | |
4 | 2, 3 | sseqtri 4018 | . . 3 ⊢ ∪ (𝐴 × 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) |
5 | 4 | unissi 4921 | . 2 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 (𝐴 ∪ 𝐵) |
6 | unipw 5456 | . 2 ⊢ ∪ 𝒫 (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵) | |
7 | 5, 6 | sseqtri 4018 | 1 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3947 ⊆ wss 3949 𝒫 cpw 4606 ∪ cuni 4912 × cxp 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-opab 5215 df-xp 5688 df-rel 5689 |
This theorem is referenced by: relfld 6284 filnetlem3 35897 |
Copyright terms: Public domain | W3C validator |