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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 4758 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | syl2anc 587 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) |
5 | wun0.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunsn 10219 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
7 | 5, 1, 2 | wunpr 10212 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
8 | 5, 6, 7 | wunpr 10212 | . 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2834 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {csn 4517 {cpr 4519 〈cop 4523 WUnicwun 10203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-v 3401 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-tr 5138 df-wun 10205 |
This theorem is referenced by: wunot 10226 wunress 16670 1strwunbndx 16706 catcoppccl 17487 catcfuccl 17488 catcxpccl 17576 |
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