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Theorem wun0 10712
Description: A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypothesis
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
Assertion
Ref Expression
wun0 (𝜑 → ∅ ∈ 𝑈)

Proof of Theorem wun0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wun0.1 . . . 4 (𝜑𝑈 ∈ WUni)
2 iswun 10698 . . . . . 6 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
32ibi 266 . . . . 5 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
43simp2d 1143 . . . 4 (𝑈 ∈ WUni → 𝑈 ≠ ∅)
51, 4syl 17 . . 3 (𝜑𝑈 ≠ ∅)
6 n0 4346 . . 3 (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥𝑈)
75, 6sylib 217 . 2 (𝜑 → ∃𝑥 𝑥𝑈)
81adantr 481 . . 3 ((𝜑𝑥𝑈) → 𝑈 ∈ WUni)
9 simpr 485 . . 3 ((𝜑𝑥𝑈) → 𝑥𝑈)
10 0ss 4396 . . . 4 ∅ ⊆ 𝑥
1110a1i 11 . . 3 ((𝜑𝑥𝑈) → ∅ ⊆ 𝑥)
128, 9, 11wunss 10706 . 2 ((𝜑𝑥𝑈) → ∅ ∈ 𝑈)
137, 12exlimddv 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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