Theorem wununi 9528

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		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		iswun 9526	. . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
4	3	ibi 256	. . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
5	4	simp3d 1075	. . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
6		simp1 1061	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈)
7	6	ralimi 2952	. . 3 ⊢ (∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥 ∈ 𝑈 ∪ 𝑥 ∈ 𝑈)
8	2, 5, 7	3syl 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∪ 𝑥 ∈ 𝑈)
9		unieq 4444	. . . 4 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
10	9	eleq1d 2686	. . 3 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝑈 ↔ ∪ 𝐴 ∈ 𝑈))
11	10	rspcv 3305	. 2 ⊢ (𝐴 ∈ 𝑈 → (∀𝑥 ∈ 𝑈 ∪ 𝑥 ∈ 𝑈 → ∪ 𝐴 ∈ 𝑈))
12	1, 8, 11	sylc 65	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∅c0 3915  𝒫 cpw 4158  {cpr 4179  ∪ cuni 4436  Tr wtr 4752  WUnicwun 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  wunun  9532  wunint  9537  wundm  9550  wunrn  9551  wunfv  9554  intwun  9557  wuncval2  9569  wunstr  15881  wunfunc  16559  wunnat  16616  catcoppccl  16758  catcfuccl  16759  catcxpccl  16847