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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 10698 | . . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
3 | 2 | ibi 266 | . . . . 5 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
4 | 3 | simp2d 1143 | . . . 4 ⊢ (𝑈 ∈ WUni → 𝑈 ≠ ∅) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ≠ ∅) |
6 | n0 4346 | . . 3 ⊢ (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑈) | |
7 | 5, 6 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝑈) |
8 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ WUni) |
9 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
10 | 0ss 4396 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ⊆ 𝑥) |
12 | 8, 9, 11 | wunss 10706 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ∈ 𝑈) |
13 | 7, 12 | exlimddv 1938 | 1 ⊢ (𝜑 → ∅ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {cpr 4630 ∪ cuni 4908 Tr wtr 5265 WUnicwun 10694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-in 3955 df-ss 3965 df-nul 4323 df-pw 4604 df-uni 4909 df-tr 5266 df-wun 10696 |
This theorem is referenced by: wunr1om 10713 wunfi 10715 wuntpos 10728 intwun 10729 r1wunlim 10731 wuncval2 10741 wunress 17194 wunressOLD 17195 catcoppccl 18066 catcoppcclOLD 18067 ex-sategoelel 34407 |
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