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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 10687 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
5 | 1, 4 | wunpw 10684 | . . 3 ⊢ (𝜑 → 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
6 | 1, 5 | wunpw 10684 | . 2 ⊢ (𝜑 → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
7 | xpsspw 5801 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) |
9 | 1, 6, 8 | wunss 10689 | 1 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∪ cun 3942 ⊆ wss 3944 𝒫 cpw 4596 × cxp 5667 WUnicwun 10677 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-opab 5204 df-tr 5259 df-xp 5675 df-rel 5676 df-wun 10679 |
This theorem is referenced by: wunpm 10702 wuncnv 10707 wunco 10710 wuntpos 10711 tskxp 10764 wuncn 11147 wunfunc 17831 wunfuncOLD 17832 wunnat 17889 wunnatOLD 17890 catcoppccl 18049 catcoppcclOLD 18050 catcfuccl 18051 catcfucclOLD 18052 catcxpccl 18141 catcxpcclOLD 18142 |
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