Theorem wununi 10746

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 4918	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2826	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝑈 ↔ ∪ 𝐴 ∈ 𝑈))
3		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
4		iswun 10744	. . . . 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 1145	. . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7		simp1 1137	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈)
8	7	ralimi 3083	. . 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 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∅c0 4333  𝒫 cpw 4600  {cpr 4628  ∪ cuni 4907  Tr wtr 5259  WUnicwun 10740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-wun 10742

This theorem is referenced by:  wunun  10750  wunint  10755  wundm  10768  wunrn  10769  wunfv  10772  intwun  10775  wuncval2  10787  wunstr  17225  wunfunc  17946  wunnat  18004  catcoppccl  18162  catcfuccl  18163  catcxpccl  18252