Theorem sspw12 4337

Description: A set is a subset of cardinal one iff it is the unit power class of some other set. (Contributed by SF, 17-Mar-2015.)

Hypothesis

Ref	Expression
sspw12.1	|- A e. _V

Assertion

Ref	Expression
sspw12	|- (A C_ 1c <-> E.x A = ~P1 x)

Distinct variable group:   x,A

Proof of Theorem sspw12

Step	Hyp	Ref	Expression
1		eqpw1uni 4331	. . 3 |- (A C_ 1c -> A = ~P1 U.A)
2		sspw12.1	. . . . 5 |- A e. _V
3	2	uniex 4318	. . . 4 |- U.A e. _V
4		pw1eq 4144	. . . . 5 |- (x = U.A -> ~P1 x = ~P1 U.A)
5	4	eqeq2d 2364	. . . 4 |- (x = U.A -> (A = ~P1 x <-> A = ~P1 U.A))
6	3, 5	spcev 2947	. . 3 |- (A = ~P1 U.A -> E.x A = ~P1 x)
7	1, 6	syl 15	. 2 |- (A C_ 1c -> E.x A = ~P1 x)
8		pw1ss1c 4159	. . . 4 |- ~P1 x C_ 1c
9		sseq1 3293	. . . 4 |- (A = ~P1 x -> (A C_ 1c <-> ~P1 x C_ 1c))
10	8, 9	mpbiri 224	. . 3 |- (A = ~P1 x -> A C_ 1c)
11	10	exlimiv 1634	. 2 |- (E.x A = ~P1 x -> A C_ 1c)
12	7, 11	impbii 180	1 |- (A C_ 1c <-> E.x A = ~P1 x)

Syntax hints:   <-> wb 176  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   C_ wss 3258  U.cuni 3892  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-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-pr 3743  df-uni 3893  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194

This theorem is referenced by:  ce0lenc1  6240