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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 4338 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | syl2anc 691 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) |
5 | wun0.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunsn 9417 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
7 | 5, 1, 2 | wunpr 9410 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
8 | 5, 6, 7 | wunpr 9410 | . 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2688 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 {cpr 4127 〈cop 4131 WUnicwun 9401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-tr 4681 df-wun 9403 |
This theorem is referenced by: wunot 9424 wunress 15767 1strwunbndx 15807 catcoppccl 16581 catcfuccl 16582 catcxpccl 16670 |
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