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Theorem iunpwss 4055

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 ℘x ⊆ ℘∪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 4008	. . 3 ⊢ (∃x ∈ A y ⊆ x → y ⊆ ∪x ∈ A x)
2		eliun 3973	. . . 4 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ∈ ℘x)
3		vex 2862	. . . . . 6 ⊢ y ∈ V
4	3	elpw 3728	. . . . 5 ⊢ (y ∈ ℘x ↔ y ⊆ x)
5	4	rexbii 2639	. . . 4 ⊢ (∃x ∈ A y ∈ ℘x ↔ ∃x ∈ A y ⊆ x)
6	2, 5	bitri 240	. . 3 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ⊆ x)
7	3	elpw 3728	. . . 4 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪A)
8		uniiun 4019	. . . . 5 ⊢ ∪A = ∪x ∈ A x
9	8	sseq2i 3296	. . . 4 ⊢ (y ⊆ ∪A ↔ y ⊆ ∪x ∈ A x)
10	7, 9	bitri 240	. . 3 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪x ∈ A x)
11	1, 6, 10	3imtr4i 257	. 2 ⊢ (y ∈ ∪x ∈ A ℘x → y ∈ ℘∪A)
12	11	ssriv 3277	1 ⊢ ∪x ∈ A ℘x ⊆ ℘∪A

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 1710 ∃wrex 2615 ⊆ wss 3257 ℘cpw 3722 ∪cuni 3891 ∪ciun 3969

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-pw 3724 df-uni 3892 df-iun 3971

This theorem is referenced by: (None)

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