Theorem iunpwss 5058

Description: Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)

Assertion

Ref	Expression
iunpwss	⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem iunpwss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssiun 4997	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
2		eliun 4949	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
3		velpw 4556	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
4	3	rexbii 3094	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
5	2, 4	bitri 275	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
6		velpw 4556	. . . 4 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
7		uniiun 5009	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
8	7	sseq2i 3964	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
9	6, 8	bitri 275	. . 3 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
10	1, 5, 9	3imtr4i 292	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴)
11	10	ssriv 3939	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 2106  ∃wrex 3071   ⊆ wss 3901  𝒫 cpw 4551  ∪ cuni 4856  ∪ ciun 4945

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-v 3444  df-in 3908  df-ss 3918  df-pw 4553  df-uni 4857  df-iun 4947

This theorem is referenced by: (None)