Theorem eqpw1uni 4331

Description: A class of singletons is equal to the unit power class of its union. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
eqpw1uni	⊢ (A ⊆ 1c → A = ℘1∪A)

Proof of Theorem eqpw1uni

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . 3 ⊢ (A ⊆ 1c → (x ∈ A → x ∈ 1c))
2		pw1ss1c 4159	. . . . 5 ⊢ ℘1∪A ⊆ 1c
3	2	sseli 3270	. . . 4 ⊢ (x ∈ ℘1∪A → x ∈ 1c)
4	3	a1i 10	. . 3 ⊢ (A ⊆ 1c → (x ∈ ℘1∪A → x ∈ 1c))
5		el1c 4140	. . . 4 ⊢ (x ∈ 1c ↔ ∃y x = {y})
6		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
7	6	snid 3761	. . . . . . . . 9 ⊢ y ∈ {y}
8		eleq2 2414	. . . . . . . . . 10 ⊢ (x = {y} → (y ∈ x ↔ y ∈ {y}))
9	8	rspcev 2956	. . . . . . . . 9 ⊢ (({y} ∈ A ∧ y ∈ {y}) → ∃x ∈ A y ∈ x)
10	7, 9	mpan2 652	. . . . . . . 8 ⊢ ({y} ∈ A → ∃x ∈ A y ∈ x)
11		el1c 4140	. . . . . . . . . . 11 ⊢ (x ∈ 1c ↔ ∃z x = {z})
12		elsn 3749	. . . . . . . . . . . . . . 15 ⊢ (y ∈ {z} ↔ y = z)
13		sneq 3745	. . . . . . . . . . . . . . . 16 ⊢ (y = z → {y} = {z})
14	13	eleq1d 2419	. . . . . . . . . . . . . . 15 ⊢ (y = z → ({y} ∈ A ↔ {z} ∈ A))
15	12, 14	sylbi 187	. . . . . . . . . . . . . 14 ⊢ (y ∈ {z} → ({y} ∈ A ↔ {z} ∈ A))
16	15	biimprcd 216	. . . . . . . . . . . . 13 ⊢ ({z} ∈ A → (y ∈ {z} → {y} ∈ A))
17		eleq1 2413	. . . . . . . . . . . . . 14 ⊢ (x = {z} → (x ∈ A ↔ {z} ∈ A))
18		eleq2 2414	. . . . . . . . . . . . . . 15 ⊢ (x = {z} → (y ∈ x ↔ y ∈ {z}))
19	18	imbi1d 308	. . . . . . . . . . . . . 14 ⊢ (x = {z} → ((y ∈ x → {y} ∈ A) ↔ (y ∈ {z} → {y} ∈ A)))
20	17, 19	imbi12d 311	. . . . . . . . . . . . 13 ⊢ (x = {z} → ((x ∈ A → (y ∈ x → {y} ∈ A)) ↔ ({z} ∈ A → (y ∈ {z} → {y} ∈ A))))
21	16, 20	mpbiri 224	. . . . . . . . . . . 12 ⊢ (x = {z} → (x ∈ A → (y ∈ x → {y} ∈ A)))
22	21	exlimiv 1634	. . . . . . . . . . 11 ⊢ (∃z x = {z} → (x ∈ A → (y ∈ x → {y} ∈ A)))
23	11, 22	sylbi 187	. . . . . . . . . 10 ⊢ (x ∈ 1c → (x ∈ A → (y ∈ x → {y} ∈ A)))
24	1, 23	syli 33	. . . . . . . . 9 ⊢ (A ⊆ 1c → (x ∈ A → (y ∈ x → {y} ∈ A)))
25	24	rexlimdv 2738	. . . . . . . 8 ⊢ (A ⊆ 1c → (∃x ∈ A y ∈ x → {y} ∈ A))
26	10, 25	impbid2 195	. . . . . . 7 ⊢ (A ⊆ 1c → ({y} ∈ A ↔ ∃x ∈ A y ∈ x))
27		eluni2 3896	. . . . . . 7 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
28	26, 27	syl6bbr 254	. . . . . 6 ⊢ (A ⊆ 1c → ({y} ∈ A ↔ y ∈ ∪A))
29		eleq1 2413	. . . . . . 7 ⊢ (x = {y} → (x ∈ A ↔ {y} ∈ A))
30		eleq1 2413	. . . . . . . 8 ⊢ (x = {y} → (x ∈ ℘1∪A ↔ {y} ∈ ℘1∪A))
31		snelpw1 4147	. . . . . . . 8 ⊢ ({y} ∈ ℘1∪A ↔ y ∈ ∪A)
32	30, 31	syl6bb 252	. . . . . . 7 ⊢ (x = {y} → (x ∈ ℘1∪A ↔ y ∈ ∪A))
33	29, 32	bibi12d 312	. . . . . 6 ⊢ (x = {y} → ((x ∈ A ↔ x ∈ ℘1∪A) ↔ ({y} ∈ A ↔ y ∈ ∪A)))
34	28, 33	syl5ibrcom 213	. . . . 5 ⊢ (A ⊆ 1c → (x = {y} → (x ∈ A ↔ x ∈ ℘1∪A)))
35	34	exlimdv 1636	. . . 4 ⊢ (A ⊆ 1c → (∃y x = {y} → (x ∈ A ↔ x ∈ ℘1∪A)))
36	5, 35	syl5bi 208	. . 3 ⊢ (A ⊆ 1c → (x ∈ 1c → (x ∈ A ↔ x ∈ ℘1∪A)))
37	1, 4, 36	pm5.21ndd 343	. 2 ⊢ (A ⊆ 1c → (x ∈ A ↔ x ∈ ℘1∪A))
38	37	eqrdv 2351	1 ⊢ (A ⊆ 1c → A = ℘1∪A)

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ⊆ wss 3258  {csn 3738  ∪cuni 3892  1_cc1c 4135  ℘₁cpw1 4136

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  ax-nin 4079  ax-sn 4088

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-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-uni 3893  df-1c 4137  df-pw1 4138

This theorem is referenced by:  pw1equn  4332  pw1eqadj  4333  sspw1  4336  sspw12  4337