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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 4599 | . 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 10666 | . . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈) |
6 | wunot.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
7 | 2, 5, 6 | wunop 10666 | . 2 ⊢ (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈) |
8 | 1, 7 | eqeltrid 2838 | 1 ⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⟨cop 4596 ⟨cotp 4598 WUnicwun 10644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4870 df-tr 5227 df-wun 10646 |
This theorem is referenced by: (None) |
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