Theorem pw111 4170

Description: The unit power class operation is one-to-one. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
pw111	|- (~P1 A = ~P1 B <-> A = B)

Proof of Theorem pw111

Dummy variables t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4111	. . . . 5 |- {x} e. _V
2		eleq1 2413	. . . . . 6 |- (t = {x} -> (t e. ~P1 A <-> {x} e. ~P1 A))
3		eleq1 2413	. . . . . 6 |- (t = {x} -> (t e. ~P1 B <-> {x} e. ~P1 B))
4	2, 3	bibi12d 312	. . . . 5 |- (t = {x} -> ((t e. ~P1 A <-> t e. ~P1 B) <-> ({x} e. ~P1 A <-> {x} e. ~P1 B)))
5	1, 4	ceqsalv 2885	. . . 4 |- (A.t(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)) <-> ({x} e. ~P1 A <-> {x} e. ~P1 B))
6		snelpw1 4146	. . . . 5 |- ({x} e. ~P1 A <-> x e. A)
7		snelpw1 4146	. . . . 5 |- ({x} e. ~P1 B <-> x e. B)
8	6, 7	bibi12i 306	. . . 4 |- (({x} e. ~P1 A <-> {x} e. ~P1 B) <-> (x e. A <-> x e. B))
9	5, 8	bitri 240	. . 3 |- (A.t(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)) <-> (x e. A <-> x e. B))
10	9	albii 1566	. 2 |- (A.xA.t(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)) <-> A.x(x e. A <-> x e. B))
11		pw1ss1c 4158	. . . 4 |- ~P1 A C_ 1c
12		pw1ss1c 4158	. . . 4 |- ~P1 B C_ 1c
13		ssofeq 4077	. . . 4 |- ((~P1 A C_ 1c /\ ~P1 B C_ 1c) -> (~P1 A = ~P1 B <-> A.t e. 1c (t e. ~P1 A <-> t e. ~P1 B)))
14	11, 12, 13	mp2an 653	. . 3 |- (~P1 A = ~P1 B <-> A.t e. 1c (t e. ~P1 A <-> t e. ~P1 B))
15		df-ral 2619	. . . 4 |- (A.t e. 1c (t e. ~P1 A <-> t e. ~P1 B) <-> A.t(t e. 1c -> (t e. ~P1 A <-> t e. ~P1 B)))
16		el1c 4139	. . . . . . . 8 |- (t e. 1c <-> E.x t = {x})
17	16	imbi1i 315	. . . . . . 7 |- ((t e. 1c -> (t e. ~P1 A <-> t e. ~P1 B)) <-> (E.x t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
18		19.23v 1891	. . . . . . 7 |- (A.x(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)) <-> (E.x t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
19	17, 18	bitr4i 243	. . . . . 6 |- ((t e. 1c -> (t e. ~P1 A <-> t e. ~P1 B)) <-> A.x(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
20	19	albii 1566	. . . . 5 |- (A.t(t e. 1c -> (t e. ~P1 A <-> t e. ~P1 B)) <-> A.tA.x(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
21		alcom 1737	. . . . 5 |- (A.tA.x(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)) <-> A.xA.t(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
22	20, 21	bitri 240	. . . 4 |- (A.t(t e. 1c -> (t e. ~P1 A <-> t e. ~P1 B)) <-> A.xA.t(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
23	15, 22	bitri 240	. . 3 |- (A.t e. 1c (t e. ~P1 A <-> t e. ~P1 B) <-> A.xA.t(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
24	14, 23	bitri 240	. 2 |- (~P1 A = ~P1 B <-> A.xA.t(t = {x} -> (t e. ~P1 A <-> t e. ~P1 B)))
25		dfcleq 2347	. 2 |- (A = B <-> A.x(x e. A <-> x e. B))
26	10, 24, 25	3bitr4i 268	1 |- (~P1 A = ~P1 B <-> A = B)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  A.wral 2614   C_ wss 3257  {csn 3737  1_cc1c 4134  ~P₁ cpw1 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-ral 2619  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:  pw1fnf1o  5855