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Mirrors > Home > MPE Home > Th. List > wunop | Structured version Visualization version GIF version |
Description: A weak universe is closed under ordered pairs. (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 |
---|---|
wunop | ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunop.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | wunop.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
3 | dfopg 4802 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) |
5 | wun0.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunsn 10472 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
7 | 5, 1, 2 | wunpr 10465 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
8 | 5, 6, 7 | wunpr 10465 | . 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2839 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 {csn 4561 {cpr 4563 〈cop 4567 WUnicwun 10456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-tr 5192 df-wun 10458 |
This theorem is referenced by: wunot 10479 1strwunbndx 16931 wunress 16960 wunressOLD 16961 catcoppccl 17832 catcoppcclOLD 17833 catcfuccl 17834 catcfucclOLD 17835 catcxpccl 17924 catcxpcclOLD 17925 |
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