Theorem iunpwss 4004

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 3954	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
2		eliun 3916	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
3		vex 2763	. . . . . 6 ⊢ 𝑦 ∈ V
4	3	elpw 3607	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
5	4	rexbii 2501	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
6	2, 5	bitri 184	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
7	3	elpw 3607	. . . 4 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
8		uniiun 3966	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
9	8	sseq2i 3206	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
10	7, 9	bitri 184	. . 3 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
11	1, 6, 10	3imtr4i 201	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴)
12	11	ssriv 3183	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 2164  ∃wrex 2473   ⊆ wss 3153  𝒫 cpw 3601  ∪ cuni 3835  ∪ ciun 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603  df-uni 3836  df-iun 3914

This theorem is referenced by: (None)