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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 10121 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
5 | 1, 4 | wunpw 10118 | . . 3 ⊢ (𝜑 → 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
6 | 1, 5 | wunpw 10118 | . 2 ⊢ (𝜑 → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
7 | xpsspw 5646 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) |
9 | 1, 6, 8 | wunss 10123 | 1 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 𝒫 cpw 4497 × cxp 5517 WUnicwun 10111 |
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-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 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-tr 5137 df-xp 5525 df-rel 5526 df-wun 10113 |
This theorem is referenced by: wunpm 10136 wuncnv 10141 wunco 10144 wuntpos 10145 tskxp 10198 wuncn 10581 wunfunc 17161 wunnat 17218 catcoppccl 17360 catcfuccl 17361 catcxpccl 17449 |
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