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Theorem wun0 9986
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 9972 . . . . . 6 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
32ibi 268 . . . . 5 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
43simp2d 1136 . . . 4 (𝑈 ∈ WUni → 𝑈 ≠ ∅)
51, 4syl 17 . . 3 (𝜑𝑈 ≠ ∅)
6 n0 4230 . . 3 (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥𝑈)
75, 6sylib 219 . 2 (𝜑 → ∃𝑥 𝑥𝑈)
81adantr 481 . . 3 ((𝜑𝑥𝑈) → 𝑈 ∈ WUni)
9 simpr 485 . . 3 ((𝜑𝑥𝑈) → 𝑥𝑈)
10 0ss 4270 . . . 4 ∅ ⊆ 𝑥
1110a1i 11 . . 3 ((𝜑𝑥𝑈) → ∅ ⊆ 𝑥)
128, 9, 11wunss 9980 . 2 ((𝜑𝑥𝑈) → ∅ ∈ 𝑈)
137, 12exlimddv 1913 1 (𝜑 → ∅ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080  wex 1761  wcel 2081  wne 2984  wral 3105  wss 3859  c0 4211  𝒫 cpw 4453  {cpr 4474   cuni 4745  Tr wtr 5063  WUnicwun 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-v 3439  df-dif 3862  df-in 3866  df-ss 3874  df-nul 4212  df-pw 4455  df-uni 4746  df-tr 5064  df-wun 9970
This theorem is referenced by:  wunr1om  9987  wunfi  9989  wuntpos  10002  intwun  10003  r1wunlim  10005  wuncval2  10015  wunress  16393  catcoppccl  17197  ex-sategoelel  32277
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