Theorem iunpwss 2039

Description: Inclusion of an indexed intersection in the power class of a union. Part of Exercise 24(b) of [Enderton] p. 33.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem iunpwss

Step	Hyp	Ref	Expression
1		ssiun 2018	. . 3 ⊢ (∃x ∈ A y ⊆ x → y ⊆ ∪x ∈ A x)
2		eliun 1998	. . . 4 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ∈ ℘x)
3		visset 1350	. . . . . 6 ⊢ y ∈ V
4	3	elpw 1801	. . . . 5 ⊢ (y ∈ ℘x ↔ y ⊆ x)
5	4	birex 1224	. . . 4 ⊢ (∃x ∈ A y ∈ ℘x ↔ ∃x ∈ A y ⊆ x)
6	2, 5	bitr 151	. . 3 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ⊆ x)
7	3	elpw 1801	. . . 4 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪A)
8		uniiun 2026	. . . . 5 ⊢ ∪A = ∪x ∈ A x
9	8	sseq2i 1525	. . . 4 ⊢ (y ⊆ ∪A ↔ y ⊆ ∪x ∈ A x)
10	7, 9	bitr 151	. . 3 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪x ∈ A x)
11	1, 6, 10	3imtr4 192	. 2 ⊢ (y ∈ ∪x ∈ A ℘x → y ∈ ℘∪A)
12	11	ssriv 1508	1 ⊢ ∪x ∈ A ℘x ⊆ ℘∪A

Syntax hints:   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  ℘cpw 1798  ∪cuni 1919  ∪ciun 1994

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799  df-uni 1920  df-iun 1996

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