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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 10134 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
5 | 1, 4 | wunpw 10131 | . . 3 ⊢ (𝜑 → 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
6 | 1, 5 | wunpw 10131 | . 2 ⊢ (𝜑 → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
7 | xpsspw 5684 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) |
9 | 1, 6, 8 | wunss 10136 | 1 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∪ cun 3936 ⊆ wss 3938 𝒫 cpw 4541 × cxp 5555 WUnicwun 10124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-opab 5131 df-tr 5175 df-xp 5563 df-rel 5564 df-wun 10126 |
This theorem is referenced by: wunpm 10149 wuncnv 10154 wunco 10157 wuntpos 10158 tskxp 10211 wuncn 10594 wunfunc 17171 wunnat 17228 catcoppccl 17370 catcfuccl 17371 catcxpccl 17459 |
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