Theorem wununi 10117

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 4811	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2874	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝑈 ↔ ∪ 𝐴 ∈ 𝑈))
3		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
4		iswun 10115	. . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
5	4	ibi 270	. . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
6	5	simp3d 1141	. . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7		simp1 1133	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈)
8	7	ralimi 3128	. . 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 3573	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∅c0 4243  𝒫 cpw 4497  {cpr 4527  ∪ cuni 4800  Tr wtr 5136  WUnicwun 10111

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137  df-wun 10113

This theorem is referenced by:  wunun  10121  wunint  10126  wundm  10139  wunrn  10140  wunfv  10143  intwun  10146  wuncval2  10158  wunstr  16499  wunfunc  17161  wunnat  17218  catcoppccl  17360  catcfuccl  17361  catcxpccl  17449