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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 4579 | . 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 10147 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) |
6 | wunot.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
7 | 2, 5, 6 | wunop 10147 | . 2 ⊢ (𝜑 → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ 𝑈) |
8 | 1, 7 | eqeltrid 2920 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵, 𝐶〉 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 〈cop 4576 〈cotp 4578 WUnicwun 10125 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-ot 4579 df-uni 4842 df-tr 5176 df-wun 10127 |
This theorem is referenced by: (None) |
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