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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 10204 | . . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
3 | 2 | ibi 270 | . . . . 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 4235 | . . 3 ⊢ (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑈) | |
7 | 5, 6 | sylib 221 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝑈) |
8 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ WUni) |
9 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
10 | 0ss 4285 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ⊆ 𝑥) |
12 | 8, 9, 11 | wunss 10212 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ∈ 𝑈) |
13 | 7, 12 | exlimddv 1942 | 1 ⊢ (𝜑 → ∅ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 ∃wex 1786 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ⊆ wss 3843 ∅c0 4211 𝒫 cpw 4488 {cpr 4518 ∪ cuni 4796 Tr wtr 5136 WUnicwun 10200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2710 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-ral 3058 df-rab 3062 df-v 3400 df-dif 3846 df-in 3850 df-ss 3860 df-nul 4212 df-pw 4490 df-uni 4797 df-tr 5137 df-wun 10202 |
This theorem is referenced by: wunr1om 10219 wunfi 10221 wuntpos 10234 intwun 10235 r1wunlim 10237 wuncval2 10247 wunress 16667 catcoppccl 17484 ex-sategoelel 32954 |
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