Theorem iunpwss 3850

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 3802	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
2		eliun 3764	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
3		vex 2644	. . . . . 6 ⊢ 𝑦 ∈ V
4	3	elpw 3463	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
5	4	rexbii 2401	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
6	2, 5	bitri 183	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
7	3	elpw 3463	. . . 4 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
8		uniiun 3813	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
9	8	sseq2i 3074	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
10	7, 9	bitri 183	. . 3 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
11	1, 6, 10	3imtr4i 200	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴)
12	11	ssriv 3051	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 1448  ∃wrex 2376   ⊆ wss 3021  𝒫 cpw 3457  ∪ cuni 3683  ∪ ciun 3760

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-uni 3684  df-iun 3762

This theorem is referenced by: (None)