Theorem eqpw1 4163

Description: A condition for equality to unit power class. (Contributed by SF, 21-Jan-2015.)

Assertion

Ref	Expression
eqpw1	|- (A = ~P1 B <-> (A C_ 1c /\ A.x({x} e. A <-> x e. B)))

Distinct variable groups:   x,A   x,B

Proof of Theorem eqpw1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw1ss1c 4159	. . 3 |- ~P1 B C_ 1c
2		sseq1 3293	. . 3 |- (A = ~P1 B -> (A C_ 1c <-> ~P1 B C_ 1c))
3	1, 2	mpbiri 224	. 2 |- (A = ~P1 B -> A C_ 1c)
4		ssofeq 4078	. . . 4 |- ((A C_ 1c /\ ~P1 B C_ 1c) -> (A = ~P1 B <-> A.y e. 1c (y e. A <-> y e. ~P1 B)))
5	1, 4	mpan2 652	. . 3 |- (A C_ 1c -> (A = ~P1 B <-> A.y e. 1c (y e. A <-> y e. ~P1 B)))
6		df-ral 2620	. . . . 5 |- (A.y e. 1c (y e. A <-> y e. ~P1 B) <-> A.y(y e. 1c -> (y e. A <-> y e. ~P1 B)))
7		el1c 4140	. . . . . . . . 9 |- (y e. 1c <-> E.x y = {x})
8	7	imbi1i 315	. . . . . . . 8 |- ((y e. 1c -> (y e. A <-> y e. ~P1 B)) <-> (E.x y = {x} -> (y e. A <-> y e. ~P1 B)))
9		19.23v 1891	. . . . . . . 8 |- (A.x(y = {x} -> (y e. A <-> y e. ~P1 B)) <-> (E.x y = {x} -> (y e. A <-> y e. ~P1 B)))
10	8, 9	bitr4i 243	. . . . . . 7 |- ((y e. 1c -> (y e. A <-> y e. ~P1 B)) <-> A.x(y = {x} -> (y e. A <-> y e. ~P1 B)))
11	10	albii 1566	. . . . . 6 |- (A.y(y e. 1c -> (y e. A <-> y e. ~P1 B)) <-> A.yA.x(y = {x} -> (y e. A <-> y e. ~P1 B)))
12		alcom 1737	. . . . . 6 |- (A.xA.y(y = {x} -> (y e. A <-> y e. ~P1 B)) <-> A.yA.x(y = {x} -> (y e. A <-> y e. ~P1 B)))
13	11, 12	bitr4i 243	. . . . 5 |- (A.y(y e. 1c -> (y e. A <-> y e. ~P1 B)) <-> A.xA.y(y = {x} -> (y e. A <-> y e. ~P1 B)))
14	6, 13	bitri 240	. . . 4 |- (A.y e. 1c (y e. A <-> y e. ~P1 B) <-> A.xA.y(y = {x} -> (y e. A <-> y e. ~P1 B)))
15		snex 4112	. . . . . . 7 |- {x} e. _V
16		eleq1 2413	. . . . . . . 8 |- (y = {x} -> (y e. A <-> {x} e. A))
17		eleq1 2413	. . . . . . . 8 |- (y = {x} -> (y e. ~P1 B <-> {x} e. ~P1 B))
18	16, 17	bibi12d 312	. . . . . . 7 |- (y = {x} -> ((y e. A <-> y e. ~P1 B) <-> ({x} e. A <-> {x} e. ~P1 B)))
19	15, 18	ceqsalv 2886	. . . . . 6 |- (A.y(y = {x} -> (y e. A <-> y e. ~P1 B)) <-> ({x} e. A <-> {x} e. ~P1 B))
20		snelpw1 4147	. . . . . . 7 |- ({x} e. ~P1 B <-> x e. B)
21	20	bibi2i 304	. . . . . 6 |- (({x} e. A <-> {x} e. ~P1 B) <-> ({x} e. A <-> x e. B))
22	19, 21	bitri 240	. . . . 5 |- (A.y(y = {x} -> (y e. A <-> y e. ~P1 B)) <-> ({x} e. A <-> x e. B))
23	22	albii 1566	. . . 4 |- (A.xA.y(y = {x} -> (y e. A <-> y e. ~P1 B)) <-> A.x({x} e. A <-> x e. B))
24	14, 23	bitri 240	. . 3 |- (A.y e. 1c (y e. A <-> y e. ~P1 B) <-> A.x({x} e. A <-> x e. B))
25	5, 24	syl6bb 252	. 2 |- (A C_ 1c -> (A = ~P1 B <-> A.x({x} e. A <-> x e. B)))
26	3, 25	biadan2 623	1 |- (A = ~P1 B <-> (A C_ 1c /\ A.x({x} e. A <-> x e. B)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  A.wral 2615   C_ wss 3258  {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-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-1c 4137  df-pw1 4138

This theorem is referenced by:  eqpw1relk  4480