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Theorem wun0 10218
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 10204 . . . . . 6 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
32ibi 270 . . . . 5 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
43simp2d 1144 . . . 4 (𝑈 ∈ WUni → 𝑈 ≠ ∅)
51, 4syl 17 . . 3 (𝜑𝑈 ≠ ∅)
6 n0 4235 . . 3 (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥𝑈)
75, 6sylib 221 . 2 (𝜑 → ∃𝑥 𝑥𝑈)
81adantr 484 . . 3 ((𝜑𝑥𝑈) → 𝑈 ∈ WUni)
9 simpr 488 . . 3 ((𝜑𝑥𝑈) → 𝑥𝑈)
10 0ss 4285 . . . 4 ∅ ⊆ 𝑥
1110a1i 11 . . 3 ((𝜑𝑥𝑈) → ∅ ⊆ 𝑥)
128, 9, 11wunss 10212 . 2 ((𝜑𝑥𝑈) → ∅ ∈ 𝑈)
137, 12exlimddv 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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