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Theorem wununi 10746
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 4918 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
21eleq1d 2826 . 2 (𝑥 = 𝐴 → ( 𝑥𝑈 𝐴𝑈))
3 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
4 iswun 10744 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
54ibi 267 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
65simp3d 1145 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
7 simp1 1137 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝑥𝑈)
87ralimi 3083 . . 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 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  c0 4333  𝒫 cpw 4600  {cpr 4628   cuni 4907  Tr wtr 5259  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-wun 10742
This theorem is referenced by:  wunun  10750  wunint  10755  wundm  10768  wunrn  10769  wunfv  10772  intwun  10775  wuncval2  10787  wunstr  17225  wunfunc  17946  wunnat  18004  catcoppccl  18162  catcfuccl  18163  catcxpccl  18252
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