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Theorem wununi 10703
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 4919 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
21eleq1d 2818 . 2 (𝑥 = 𝐴 → ( 𝑥𝑈 𝐴𝑈))
3 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
4 iswun 10701 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
54ibi 266 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
65simp3d 1144 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
7 simp1 1136 . . . 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 3613 1 (𝜑 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  c0 4322  𝒫 cpw 4602  {cpr 4630   cuni 4908  Tr wtr 5265  WUnicwun 10697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965  df-uni 4909  df-tr 5266  df-wun 10699
This theorem is referenced by:  wunun  10707  wunint  10712  wundm  10725  wunrn  10726  wunfv  10729  intwun  10732  wuncval2  10744  wunstr  17125  wunfunc  17853  wunfuncOLD  17854  wunnat  17911  wunnatOLD  17912  catcoppccl  18071  catcoppcclOLD  18072  catcfuccl  18073  catcfucclOLD  18074  catcxpccl  18163  catcxpcclOLD  18164
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