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| Mirrors > Home > MPE Home > Th. List > wunot | Structured version Visualization version GIF version | ||
| Description: A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
| wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| wunop.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
| wunot.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑈) |
| Ref | Expression |
|---|---|
| wunot | ⊢ (𝜑 → 〈𝐴, 𝐵, 𝐶〉 ∈ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ot 4635 | . 2 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
| 2 | wun0.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
| 3 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 4 | wunop.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
| 5 | 2, 3, 4 | wunop 10762 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) |
| 6 | wunot.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
| 7 | 2, 5, 6 | wunop 10762 | . 2 ⊢ (𝜑 → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ 𝑈) |
| 8 | 1, 7 | eqeltrid 2845 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵, 𝐶〉 ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 〈cop 4632 〈cotp 4634 WUnicwun 10740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-tr 5260 df-wun 10742 |
| This theorem is referenced by: (None) |
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