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Theorem wununi 10120
Description: A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wununi (𝜑 𝐴𝑈)

Proof of Theorem wununi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4838 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
21eleq1d 2895 . 2 (𝑥 = 𝐴 → ( 𝑥𝑈 𝐴𝑈))
3 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
4 iswun 10118 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
54ibi 269 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
65simp3d 1138 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
7 simp1 1130 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝑥𝑈)
87ralimi 3158 . . 3 (∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥𝑈 𝑥𝑈)
93, 6, 83syl 18 . 2 (𝜑 → ∀𝑥𝑈 𝑥𝑈)
10 wununi.2 . 2 (𝜑𝐴𝑈)
112, 9, 10rspcdva 3623 1 (𝜑 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wne 3014  wral 3136  c0 4289  𝒫 cpw 4537  {cpr 4561   cuni 4830  Tr wtr 5163  WUnicwun 10114
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-in 3941  df-ss 3950  df-uni 4831  df-tr 5164  df-wun 10116
This theorem is referenced by:  wunun  10124  wunint  10129  wundm  10142  wunrn  10143  wunfv  10146  intwun  10149  wuncval2  10161  wunstr  16499  wunfunc  17161  wunnat  17218  catcoppccl  17360  catcfuccl  17361  catcxpccl  17449
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