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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 4864 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | syl2anc 583 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) |
5 | wun0.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunsn 10708 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
7 | 5, 1, 2 | wunpr 10701 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
8 | 5, 6, 7 | wunpr 10701 | . 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2825 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4621 {cpr 4623 〈cop 4627 WUnicwun 10692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-tr 5257 df-wun 10694 |
This theorem is referenced by: wunot 10715 1strwunbndx 17164 wunress 17196 wunressOLD 17197 catcoppccl 18071 catcoppcclOLD 18072 catcfuccl 18073 catcfucclOLD 18074 catcxpccl 18163 catcxpcclOLD 18164 |
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