Theorem iunpwss 4056

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 4009	. . 3 ⊢ (∃x ∈ A y ⊆ x → y ⊆ ∪x ∈ A x)
2		eliun 3974	. . . 4 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ∈ ℘x)
3		vex 2863	. . . . . 6 ⊢ y ∈ V
4	3	elpw 3729	. . . . 5 ⊢ (y ∈ ℘x ↔ y ⊆ x)
5	4	rexbii 2640	. . . 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 3729	. . . 4 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪A)
8		uniiun 4020	. . . . 5 ⊢ ∪A = ∪x ∈ A x
9	8	sseq2i 3297	. . . 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 3278	1 ⊢ ∪x ∈ A ℘x ⊆ ℘∪A

Syntax hints:   ∈ wcel 1710  ∃wrex 2616   ⊆ wss 3258  ℘cpw 3723  ∪cuni 3892  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725  df-uni 3893  df-iun 3972

This theorem is referenced by: (None)