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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 9532 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
5 | 1, 4 | wunpw 9529 | . . 3 ⊢ (𝜑 → 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
6 | 1, 5 | wunpw 9529 | . 2 ⊢ (𝜑 → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) |
7 | xpsspw 5233 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) |
9 | 1, 6, 8 | wunss 9534 | 1 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 𝒫 cpw 4158 × cxp 5112 WUnicwun 9522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-opab 4713 df-tr 4753 df-xp 5120 df-rel 5121 df-wun 9524 |
This theorem is referenced by: wunpm 9547 wuncnv 9552 wunco 9555 wuntpos 9556 tskxp 9609 wuncn 9991 wunfunc 16559 wunnat 16616 catcoppccl 16758 catcfuccl 16759 catcxpccl 16847 |
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