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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 10694 | . . . . . 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 1140 | . . . 4 ⊢ (𝑈 ∈ WUni → 𝑈 ≠ ∅) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ≠ ∅) |
6 | n0 4338 | . . 3 ⊢ (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑈) | |
7 | 5, 6 | sylib 217 | . 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 4388 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ⊆ 𝑥) |
12 | 8, 9, 11 | wunss 10702 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ∈ 𝑈) |
13 | 7, 12 | exlimddv 1930 | 1 ⊢ (𝜑 → ∅ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⊆ wss 3940 ∅c0 4314 𝒫 cpw 4594 {cpr 4622 ∪ cuni 4899 Tr wtr 5255 WUnicwun 10690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-in 3947 df-ss 3957 df-nul 4315 df-pw 4596 df-uni 4900 df-tr 5256 df-wun 10692 |
This theorem is referenced by: wunr1om 10709 wunfi 10711 wuntpos 10724 intwun 10725 r1wunlim 10727 wuncval2 10737 wunress 17193 wunressOLD 17194 catcoppccl 18068 catcoppcclOLD 18069 ex-sategoelel 34867 |
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