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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 9972 | . . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
3 | 2 | ibi 268 | . . . . 5 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
4 | 3 | simp2d 1136 | . . . 4 ⊢ (𝑈 ∈ WUni → 𝑈 ≠ ∅) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ≠ ∅) |
6 | n0 4230 | . . 3 ⊢ (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑈) | |
7 | 5, 6 | sylib 219 | . 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 4270 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ⊆ 𝑥) |
12 | 8, 9, 11 | wunss 9980 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∅ ∈ 𝑈) |
13 | 7, 12 | exlimddv 1913 | 1 ⊢ (𝜑 → ∅ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 ∃wex 1761 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ⊆ wss 3859 ∅c0 4211 𝒫 cpw 4453 {cpr 4474 ∪ cuni 4745 Tr wtr 5063 WUnicwun 9968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-v 3439 df-dif 3862 df-in 3866 df-ss 3874 df-nul 4212 df-pw 4455 df-uni 4746 df-tr 5064 df-wun 9970 |
This theorem is referenced by: wunr1om 9987 wunfi 9989 wuntpos 10002 intwun 10003 r1wunlim 10005 wuncval2 10015 wunress 16393 catcoppccl 17197 ex-sategoelel 32277 |
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