Theorem elpw1 4144

Description: Membership in a unit power class. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
elpw1	⊢ (A ∈ ℘1B ↔ ∃x ∈ B A = {x})

Distinct variable groups:   x,A   x,B

Proof of Theorem elpw1

Step	Hyp	Ref	Expression
1		df-pw1 4137	. . . 4 ⊢ ℘1B = (℘B ∩ 1c)
2	1	eleq2i 2417	. . 3 ⊢ (A ∈ ℘1B ↔ A ∈ (℘B ∩ 1c))
3		elin 3219	. . 3 ⊢ (A ∈ (℘B ∩ 1c) ↔ (A ∈ ℘B ∧ A ∈ 1c))
4	2, 3	bitri 240	. 2 ⊢ (A ∈ ℘1B ↔ (A ∈ ℘B ∧ A ∈ 1c))
5		el1c 4139	. . . . 5 ⊢ (A ∈ 1c ↔ ∃x A = {x})
6	5	anbi2i 675	. . . 4 ⊢ ((A ∈ ℘B ∧ A ∈ 1c) ↔ (A ∈ ℘B ∧ ∃x A = {x}))
7		19.42v 1905	. . . 4 ⊢ (∃x(A ∈ ℘B ∧ A = {x}) ↔ (A ∈ ℘B ∧ ∃x A = {x}))
8	6, 7	bitr4i 243	. . 3 ⊢ ((A ∈ ℘B ∧ A ∈ 1c) ↔ ∃x(A ∈ ℘B ∧ A = {x}))
9		eleq1 2413	. . . . . . 7 ⊢ (A = {x} → (A ∈ ℘B ↔ {x} ∈ ℘B))
10		snex 4111	. . . . . . . . 9 ⊢ {x} ∈ V
11	10	elpw 3728	. . . . . . . 8 ⊢ ({x} ∈ ℘B ↔ {x} ⊆ B)
12		vex 2862	. . . . . . . . 9 ⊢ x ∈ V
13	12	snss 3838	. . . . . . . 8 ⊢ (x ∈ B ↔ {x} ⊆ B)
14	11, 13	bitr4i 243	. . . . . . 7 ⊢ ({x} ∈ ℘B ↔ x ∈ B)
15	9, 14	syl6bb 252	. . . . . 6 ⊢ (A = {x} → (A ∈ ℘B ↔ x ∈ B))
16	15	pm5.32ri 619	. . . . 5 ⊢ ((A ∈ ℘B ∧ A = {x}) ↔ (x ∈ B ∧ A = {x}))
17	16	exbii 1582	. . . 4 ⊢ (∃x(A ∈ ℘B ∧ A = {x}) ↔ ∃x(x ∈ B ∧ A = {x}))
18		df-rex 2620	. . . 4 ⊢ (∃x ∈ B A = {x} ↔ ∃x(x ∈ B ∧ A = {x}))
19	17, 18	bitr4i 243	. . 3 ⊢ (∃x(A ∈ ℘B ∧ A = {x}) ↔ ∃x ∈ B A = {x})
20	8, 19	bitri 240	. 2 ⊢ ((A ∈ ℘B ∧ A ∈ 1c) ↔ ∃x ∈ B A = {x})
21	4, 20	bitri 240	1 ⊢ (A ∈ ℘1B ↔ ∃x ∈ B A = {x})

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∩ cin 3208   ⊆ wss 3257  ℘cpw 3722  {csn 3737  1_cc1c 4134  ℘₁cpw1 4135

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 4078  ax-sn 4087

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-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137

This theorem is referenced by:  elpw12  4145  snelpw1  4146  elpw11c  4147  elpw121c  4148  elpw131c  4149  elpw141c  4150  elpw151c  4151  elpw161c  4152  elpw171c  4153  elpw181c  4154  elpw191c  4155  elpw1101c  4156  elpw1111c  4157  pw1un  4163  pw1in  4164  pw1sn  4165  pw1disj  4167  df1c2  4168  dfpw12  4301  unipw1  4325  vfinspss  4551  vfinncsp  4554  elimapw1  4944  elimapw12  4945  elimapw13  4946  dmsi  5519  pw1fnex  5852  enpw1  6062  enpw1pw  6075  nenpw1pwlem2  6085  scancan  6331