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Theorem wun0 10474
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 10460 . . . . . 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 1142 . . . 4 (𝑈 ∈ WUni → 𝑈 ≠ ∅)
51, 4syl 17 . . 3 (𝜑𝑈 ≠ ∅)
6 n0 4280 . . 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 4330 . . . 4 ∅ ⊆ 𝑥
1110a1i 11 . . 3 ((𝜑𝑥𝑈) → ∅ ⊆ 𝑥)
128, 9, 11wunss 10468 . 2 ((𝜑𝑥𝑈) → ∅ ∈ 𝑈)
137, 12exlimddv 1938 1 (𝜑 → ∅ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086  wex 1782  wcel 2106  wne 2943  wral 3064  wss 3887  c0 4256  𝒫 cpw 4533  {cpr 4563   cuni 4839  Tr wtr 5191  WUnicwun 10456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-uni 4840  df-tr 5192  df-wun 10458
This theorem is referenced by:  wunr1om  10475  wunfi  10477  wuntpos  10490  intwun  10491  r1wunlim  10493  wuncval2  10503  wunress  16960  wunressOLD  16961  catcoppccl  17832  catcoppcclOLD  17833  ex-sategoelel  33383
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