Theorem sspw1 4336

Description: A condition for being a subclass of a unit power class. Corollary 2 of theorem IX.6.14 of [Rosser] p. 255. (Contributed by SF, 3-Feb-2015.)

Hypothesis

Ref	Expression
sspw1.1	⊢ A ∈ V

Assertion

Ref	Expression
sspw1	⊢ (A ⊆ ℘1B ↔ ∃x(x ⊆ B ∧ A = ℘1x))

Distinct variable groups:   x,A   x,B

Proof of Theorem sspw1

Step	Hyp	Ref	Expression
1		uniss 3913	. . . 4 ⊢ (A ⊆ ℘1B → ∪A ⊆ ∪℘1B)
2		unipw1 4326	. . . 4 ⊢ ∪℘1B = B
3	1, 2	syl6sseq 3318	. . 3 ⊢ (A ⊆ ℘1B → ∪A ⊆ B)
4		pw1ss1c 4159	. . . . 5 ⊢ ℘1B ⊆ 1c
5		sstr 3281	. . . . 5 ⊢ ((A ⊆ ℘1B ∧ ℘1B ⊆ 1c) → A ⊆ 1c)
6	4, 5	mpan2 652	. . . 4 ⊢ (A ⊆ ℘1B → A ⊆ 1c)
7		eqpw1uni 4331	. . . 4 ⊢ (A ⊆ 1c → A = ℘1∪A)
8	6, 7	syl 15	. . 3 ⊢ (A ⊆ ℘1B → A = ℘1∪A)
9		sspw1.1	. . . . 5 ⊢ A ∈ V
10	9	uniex 4318	. . . 4 ⊢ ∪A ∈ V
11		sseq1 3293	. . . . 5 ⊢ (x = ∪A → (x ⊆ B ↔ ∪A ⊆ B))
12		pw1eq 4144	. . . . . 6 ⊢ (x = ∪A → ℘1x = ℘1∪A)
13	12	eqeq2d 2364	. . . . 5 ⊢ (x = ∪A → (A = ℘1x ↔ A = ℘1∪A))
14	11, 13	anbi12d 691	. . . 4 ⊢ (x = ∪A → ((x ⊆ B ∧ A = ℘1x) ↔ (∪A ⊆ B ∧ A = ℘1∪A)))
15	10, 14	spcev 2947	. . 3 ⊢ ((∪A ⊆ B ∧ A = ℘1∪A) → ∃x(x ⊆ B ∧ A = ℘1x))
16	3, 8, 15	syl2anc 642	. 2 ⊢ (A ⊆ ℘1B → ∃x(x ⊆ B ∧ A = ℘1x))
17		pw1ss 4170	. . . . 5 ⊢ (x ⊆ B → ℘1x ⊆ ℘1B)
18		sseq1 3293	. . . . 5 ⊢ (A = ℘1x → (A ⊆ ℘1B ↔ ℘1x ⊆ ℘1B))
19	17, 18	syl5ibr 212	. . . 4 ⊢ (A = ℘1x → (x ⊆ B → A ⊆ ℘1B))
20	19	impcom 419	. . 3 ⊢ ((x ⊆ B ∧ A = ℘1x) → A ⊆ ℘1B)
21	20	exlimiv 1634	. 2 ⊢ (∃x(x ⊆ B ∧ A = ℘1x) → A ⊆ ℘1B)
22	16, 21	impbii 180	1 ⊢ (A ⊆ ℘1B ↔ ∃x(x ⊆ B ∧ A = ℘1x))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  ∪cuni 3892  1_cc1c 4135  ℘₁cpw1 4136

This proof depends on 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  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194

This theorem is used by:  vfinspsslem1  4551  pw1fnf1o  5856  enpw1pw  6076  ce0addcnnul  6180