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Mirrors > Home > MPE Home > Th. List > wun0 | Structured version Visualization version GIF version |
Description: A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
Ref | Expression |
---|---|
wun0 | ⊢ (𝜑 → ∅ ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | iswun 10742 | . . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
3 | 2 | ibi 267 | . . . . 5 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
4 | 3 | simp2d 1142 | . . . 4 ⊢ (𝑈 ∈ WUni → 𝑈 ≠ ∅) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ≠ ∅) |
6 | n0 4359 | . . 3 ⊢ (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑈) | |
7 | 5, 6 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝑈) |
8 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ WUni) |
9 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
10 | 0ss 4406 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ⊆ 𝑥) |
12 | 8, 9, 11 | wunss 10750 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ∈ 𝑈) |
13 | 7, 12 | exlimddv 1933 | 1 ⊢ (𝜑 → ∅ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {cpr 4633 ∪ cuni 4912 Tr wtr 5265 WUnicwun 10738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-uni 4913 df-tr 5266 df-wun 10740 |
This theorem is referenced by: wunr1om 10757 wunfi 10759 wuntpos 10772 intwun 10773 r1wunlim 10775 wuncval2 10785 wunress 17296 wunressOLD 17297 catcoppccl 18171 catcoppcclOLD 18172 ex-sategoelel 35406 |
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