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Mirrors > Home > MPE Home > Th. List > wunxp | Structured version Visualization version GIF version |
Description: A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunop.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Ref | Expression |
---|---|
wunxp | ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | wunop.3 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
4 | 1, 2, 3 | wunun 9411 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
5 | 1, 4 | wunpw 9408 | . . 3 ⊢ (𝜑 → 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
6 | 1, 5 | wunpw 9408 | . 2 ⊢ (𝜑 → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
7 | xpsspw 5156 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) |
9 | 1, 6, 8 | wunss 9413 | 1 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 × cxp 5036 WUnicwun 9401 |
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-ne 2782 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-tr 4681 df-xp 5044 df-rel 5045 df-wun 9403 |
This theorem is referenced by: wunpm 9426 wuncnv 9431 wunco 9434 wuntpos 9435 tskxp 9488 wuncn 9870 wunfunc 16382 wunnat 16439 catcoppccl 16581 catcfuccl 16582 catcxpccl 16670 |
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