Theorem elpw1 4145

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

Assertion

Ref	Expression
elpw1	|- (A e. ~P1 B <-> E.x e. B A = {x})

Distinct variable groups:   x,A   x,B

Proof of Theorem elpw1

Step	Hyp	Ref	Expression
1		df-pw1 4138	. . . 4 |- ~P1 B = (~PB i^i 1c)
2	1	eleq2i 2417	. . 3 |- (A e. ~P1 B <-> A e. (~PB i^i 1c))
3		elin 3220	. . 3 |- (A e. (~PB i^i 1c) <-> (A e. ~PB /\ A e. 1c))
4	2, 3	bitri 240	. 2 |- (A e. ~P1 B <-> (A e. ~PB /\ A e. 1c))
5		el1c 4140	. . . . 5 |- (A e. 1c <-> E.x A = {x})
6	5	anbi2i 675	. . . 4 |- ((A e. ~PB /\ A e. 1c) <-> (A e. ~PB /\ E.x A = {x}))
7		19.42v 1905	. . . 4 |- (E.x(A e. ~PB /\ A = {x}) <-> (A e. ~PB /\ E.x A = {x}))
8	6, 7	bitr4i 243	. . 3 |- ((A e. ~PB /\ A e. 1c) <-> E.x(A e. ~PB /\ A = {x}))
9		eleq1 2413	. . . . . . 7 |- (A = {x} -> (A e. ~PB <-> {x} e. ~PB))
10		snex 4112	. . . . . . . . 9 |- {x} e. _V
11	10	elpw 3729	. . . . . . . 8 |- ({x} e. ~PB <-> {x} C_ B)
12		vex 2863	. . . . . . . . 9 |- x e. _V
13	12	snss 3839	. . . . . . . 8 |- (x e. B <-> {x} C_ B)
14	11, 13	bitr4i 243	. . . . . . 7 |- ({x} e. ~PB <-> x e. B)
15	9, 14	syl6bb 252	. . . . . 6 |- (A = {x} -> (A e. ~PB <-> x e. B))
16	15	pm5.32ri 619	. . . . 5 |- ((A e. ~PB /\ A = {x}) <-> (x e. B /\ A = {x}))
17	16	exbii 1582	. . . 4 |- (E.x(A e. ~PB /\ A = {x}) <-> E.x(x e. B /\ A = {x}))
18		df-rex 2621	. . . 4 |- (E.x e. B A = {x} <-> E.x(x e. B /\ A = {x}))
19	17, 18	bitr4i 243	. . 3 |- (E.x(A e. ~PB /\ A = {x}) <-> E.x e. B A = {x})
20	8, 19	bitri 240	. 2 |- ((A e. ~PB /\ A e. 1c) <-> E.x e. B A = {x})
21	4, 20	bitri 240	1 |- (A e. ~P1 B <-> E.x e. B A = {x})

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616   i^i cin 3209   C_ wss 3258  ~Pcpw 3723  {csn 3738  1_cc1c 4135  ~P₁ cpw1 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-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-1c 4137  df-pw1 4138

This theorem is referenced by:  elpw12  4146  snelpw1  4147  elpw11c  4148  elpw121c  4149  elpw131c  4150  elpw141c  4151  elpw151c  4152  elpw161c  4153  elpw171c  4154  elpw181c  4155  elpw191c  4156  elpw1101c  4157  elpw1111c  4158  pw1un  4164  pw1in  4165  pw1sn  4166  pw1disj  4168  df1c2  4169  dfpw12  4302  unipw1  4326  vfinspss  4552  vfinncsp  4555  elimapw1  4945  elimapw12  4946  elimapw13  4947  dmsi  5520  pw1fnex  5853  enpw1  6063  enpw1pw  6076  nenpw1pwlem2  6086  scancan  6332