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Theorem wununi 9473
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 wununi.2 . 2 (𝜑𝐴𝑈)
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 iswun 9471 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
43ibi 256 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
54simp3d 1073 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
6 simp1 1059 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝑥𝑈)
76ralimi 2952 . . 3 (∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥𝑈 𝑥𝑈)
82, 5, 73syl 18 . 2 (𝜑 → ∀𝑥𝑈 𝑥𝑈)
9 unieq 4415 . . . 4 (𝑥 = 𝐴 𝑥 = 𝐴)
109eleq1d 2688 . . 3 (𝑥 = 𝐴 → ( 𝑥𝑈 𝐴𝑈))
1110rspcv 3296 . 2 (𝐴𝑈 → (∀𝑥𝑈 𝑥𝑈 𝐴𝑈))
121, 8, 11sylc 65 1 (𝜑 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  c0 3896  𝒫 cpw 4135  {cpr 4155   cuni 4407  Tr wtr 4717  WUnicwun 9467
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-v 3193  df-in 3567  df-ss 3574  df-uni 4408  df-tr 4718  df-wun 9469
This theorem is referenced by:  wunun  9477  wunint  9482  wundm  9495  wunrn  9496  wunfv  9499  intwun  9502  wuncval2  9514  wunstr  15798  wunfunc  16475  wunnat  16532  catcoppccl  16674  catcfuccl  16675  catcxpccl  16763
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