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Theorem wun0 10708
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 10694 . . . . . 6 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
32ibi 267 . . . . 5 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
43simp2d 1140 . . . 4 (𝑈 ∈ WUni → 𝑈 ≠ ∅)
51, 4syl 17 . . 3 (𝜑𝑈 ≠ ∅)
6 n0 4338 . . 3 (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥𝑈)
75, 6sylib 217 . 2 (𝜑 → ∃𝑥 𝑥𝑈)
81adantr 480 . . 3 ((𝜑𝑥𝑈) → 𝑈 ∈ WUni)
9 simpr 484 . . 3 ((𝜑𝑥𝑈) → 𝑥𝑈)
10 0ss 4388 . . . 4 ∅ ⊆ 𝑥
1110a1i 11 . . 3 ((𝜑𝑥𝑈) → ∅ ⊆ 𝑥)
128, 9, 11wunss 10702 . 2 ((𝜑𝑥𝑈) → ∅ ∈ 𝑈)
137, 12exlimddv 1930 1 (𝜑 → ∅ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wex 1773  wcel 2098  wne 2932  wral 3053  wss 3940  c0 4314  𝒫 cpw 4594  {cpr 4622   cuni 4899  Tr wtr 5255  WUnicwun 10690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-in 3947  df-ss 3957  df-nul 4315  df-pw 4596  df-uni 4900  df-tr 5256  df-wun 10692
This theorem is referenced by:  wunr1om  10709  wunfi  10711  wuntpos  10724  intwun  10725  r1wunlim  10727  wuncval2  10737  wunress  17193  wunressOLD  17194  catcoppccl  18068  catcoppcclOLD  18069  ex-sategoelel  34867
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