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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 10744 | . . . . . 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 1144 | . . . 4 ⊢ (𝑈 ∈ WUni → 𝑈 ≠ ∅) |
| 5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ≠ ∅) |
| 6 | n0 4353 | . . 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 4400 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
| 11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ⊆ 𝑥) |
| 12 | 8, 9, 11 | wunss 10752 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ∈ 𝑈) |
| 13 | 7, 12 | exlimddv 1935 | 1 ⊢ (𝜑 → ∅ ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {cpr 4628 ∪ cuni 4907 Tr wtr 5259 WUnicwun 10740 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-uni 4908 df-tr 5260 df-wun 10742 |
| This theorem is referenced by: wunr1om 10759 wunfi 10761 wuntpos 10774 intwun 10775 r1wunlim 10777 wuncval2 10787 wunress 17295 wunressOLD 17296 catcoppccl 18162 ex-sategoelel 35426 |
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