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Theorem wununi 9712
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 9710 . . . . 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 1138 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
6 simp1 1130 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝑥𝑈)
76ralimi 3082 . . 3 (∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥𝑈 𝑥𝑈)
82, 5, 73syl 18 . 2 (𝜑 → ∀𝑥𝑈 𝑥𝑈)
9 unieq 4588 . . . 4 (𝑥 = 𝐴 𝑥 = 𝐴)
109eleq1d 2816 . . 3 (𝑥 = 𝐴 → ( 𝑥𝑈 𝐴𝑈))
1110rspcv 3437 . 2 (𝐴𝑈 → (∀𝑥𝑈 𝑥𝑈 𝐴𝑈))
121, 8, 11sylc 65 1 (𝜑 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  c0 4050  𝒫 cpw 4294  {cpr 4315   cuni 4580  Tr wtr 4896  WUnicwun 9706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-v 3334  df-in 3714  df-ss 3721  df-uni 4581  df-tr 4897  df-wun 9708
This theorem is referenced by:  wunun  9716  wunint  9721  wundm  9734  wunrn  9735  wunfv  9738  intwun  9741  wuncval2  9753  wunstr  16075  wunfunc  16752  wunnat  16809  catcoppccl  16951  catcfuccl  16952  catcxpccl  17040
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