Theorem sspw1 4336

Description: A condition for being a subclass of a unit power class. Corollary 2 of theorem IX.6.14 of [Rosser] p. 255. (Contributed by SF, 3-Feb-2015.)

Hypothesis

Ref	Expression
sspw1.1	|- A e. _V

Assertion

Ref	Expression
sspw1	|- (A C_ ~P1 B <-> E.x(x C_ B /\ A = ~P1 x))

Distinct variable groups:   x,A   x,B

Proof of Theorem sspw1

Step	Hyp	Ref	Expression
1		uniss 3913	. . . 4 |- (A C_ ~P1 B -> U.A C_ U.~P1 B)
2		unipw1 4326	. . . 4 |- U.~P1 B = B
3	1, 2	syl6sseq 3318	. . 3 |- (A C_ ~P1 B -> U.A C_ B)
4		pw1ss1c 4159	. . . . 5 |- ~P1 B C_ 1c
5		sstr 3281	. . . . 5 |- ((A C_ ~P1 B /\ ~P1 B C_ 1c) -> A C_ 1c)
6	4, 5	mpan2 652	. . . 4 |- (A C_ ~P1 B -> A C_ 1c)
7		eqpw1uni 4331	. . . 4 |- (A C_ 1c -> A = ~P1 U.A)
8	6, 7	syl 15	. . 3 |- (A C_ ~P1 B -> A = ~P1 U.A)
9		sspw1.1	. . . . 5 |- A e. _V
10	9	uniex 4318	. . . 4 |- U.A e. _V
11		sseq1 3293	. . . . 5 |- (x = U.A -> (x C_ B <-> U.A C_ B))
12		pw1eq 4144	. . . . . 6 |- (x = U.A -> ~P1 x = ~P1 U.A)
13	12	eqeq2d 2364	. . . . 5 |- (x = U.A -> (A = ~P1 x <-> A = ~P1 U.A))
14	11, 13	anbi12d 691	. . . 4 |- (x = U.A -> ((x C_ B /\ A = ~P1 x) <-> (U.A C_ B /\ A = ~P1 U.A)))
15	10, 14	spcev 2947	. . 3 |- ((U.A C_ B /\ A = ~P1 U.A) -> E.x(x C_ B /\ A = ~P1 x))
16	3, 8, 15	syl2anc 642	. 2 |- (A C_ ~P1 B -> E.x(x C_ B /\ A = ~P1 x))
17		pw1ss 4170	. . . . 5 |- (x C_ B -> ~P1 x C_ ~P1 B)
18		sseq1 3293	. . . . 5 |- (A = ~P1 x -> (A C_ ~P1 B <-> ~P1 x C_ ~P1 B))
19	17, 18	syl5ibr 212	. . . 4 |- (A = ~P1 x -> (x C_ B -> A C_ ~P1 B))
20	19	impcom 419	. . 3 |- ((x C_ B /\ A = ~P1 x) -> A C_ ~P1 B)
21	20	exlimiv 1634	. 2 |- (E.x(x C_ B /\ A = ~P1 x) -> A C_ ~P1 B)
22	16, 21	impbii 180	1 |- (A C_ ~P1 B <-> E.x(x C_ B /\ A = ~P1 x))

Syntax hints:   <-> wb 176   /\ wa 358  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:  vfinspsslem1  4551  pw1fnf1o  5856  enpw1pw  6076  ce0addcnnul  6180