Theorem wununi 10744

Description: A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wununi	⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)

Proof of Theorem wununi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4923	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2824	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝑈 ↔ ∪ 𝐴 ∈ 𝑈))
3		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
4		iswun 10742	. . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
5	4	ibi 267	. . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
6	5	simp3d 1143	. . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7		simp1 1135	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈)
8	7	ralimi 3081	. . 3 ⊢ (∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥 ∈ 𝑈 ∪ 𝑥 ∈ 𝑈)
9	3, 6, 8	3syl 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∪ 𝑥 ∈ 𝑈)
10		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
11	2, 9, 10	rspcdva 3623	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∅c0 4339  𝒫 cpw 4605  {cpr 4633  ∪ cuni 4912  Tr wtr 5265  WUnicwun 10738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-uni 4913  df-tr 5266  df-wun 10740

This theorem is referenced by:  wunun  10748  wunint  10753  wundm  10766  wunrn  10767  wunfv  10770  intwun  10773  wuncval2  10785  wunstr  17222  wunfunc  17952  wunfuncOLD  17953  wunnat  18011  wunnatOLD  18012  catcoppccl  18171  catcoppcclOLD  18172  catcfuccl  18173  catcfucclOLD  18174  catcxpccl  18263  catcxpcclOLD  18264