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Theorem wun0 9487
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 9473 . . . . . 6 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
32ibi 256 . . . . 5 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
43simp2d 1072 . . . 4 (𝑈 ∈ WUni → 𝑈 ≠ ∅)
51, 4syl 17 . . 3 (𝜑𝑈 ≠ ∅)
6 n0 3909 . . 3 (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥𝑈)
75, 6sylib 208 . 2 (𝜑 → ∃𝑥 𝑥𝑈)
81adantr 481 . . 3 ((𝜑𝑥𝑈) → 𝑈 ∈ WUni)
9 simpr 477 . . 3 ((𝜑𝑥𝑈) → 𝑥𝑈)
10 0ss 3946 . . . 4 ∅ ⊆ 𝑥
1110a1i 11 . . 3 ((𝜑𝑥𝑈) → ∅ ⊆ 𝑥)
128, 9, 11wunss 9481 . 2 ((𝜑𝑥𝑈) → ∅ ∈ 𝑈)
137, 12exlimddv 1860 1 (𝜑 → ∅ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wex 1701  wcel 1987  wne 2790  wral 2907  wss 3556  c0 3893  𝒫 cpw 4132  {cpr 4152   cuni 4404  Tr wtr 4714  WUnicwun 9469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3559  df-in 3563  df-ss 3570  df-nul 3894  df-pw 4134  df-uni 4405  df-tr 4715  df-wun 9471
This theorem is referenced by:  wunr1om  9488  wunfi  9490  wuntpos  9503  intwun  9504  r1wunlim  9506  wuncval2  9516  wunress  15864  catcoppccl  16682
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