Theorem sfin01 4528

Description: Zero and one satisfy S_fin. Theorem X.1.42 of [Rosser] p. 530. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
sfin01	|- Sfin (0c, 1c)

Proof of Theorem sfin01

Step	Hyp	Ref	Expression
1		peano1 4402	. 2 |- 0c e. Nn
2		1cnnc 4408	. 2 |- 1c e. Nn
3		pw10 4161	. . 3 |- ~P1 (/) = (/)
4		0ex 4110	. . . 4 |- (/) e. _V
5	4	snel1c 4140	. . 3 |- {(/)} e. 1c
6		el0c 4421	. . . . . 6 |- (~P1 a e. 0c <-> ~P1 a = (/))
7		pw1eq 4143	. . . . . . 7 |- (a = (/) -> ~P1 a = ~P1 (/))
8	7	eqeq1d 2361	. . . . . 6 |- (a = (/) -> (~P1 a = (/) <-> ~P1 (/) = (/)))
9	6, 8	syl5bb 248	. . . . 5 |- (a = (/) -> (~P1 a e. 0c <-> ~P1 (/) = (/)))
10		pweq 3725	. . . . . . 7 |- (a = (/) -> ~Pa = ~P(/))
11		pw0 4160	. . . . . . 7 |- ~P(/) = {(/)}
12	10, 11	syl6eq 2401	. . . . . 6 |- (a = (/) -> ~Pa = {(/)})
13	12	eleq1d 2419	. . . . 5 |- (a = (/) -> (~Pa e. 1c <-> {(/)} e. 1c))
14	9, 13	anbi12d 691	. . . 4 |- (a = (/) -> ((~P1 a e. 0c /\ ~Pa e. 1c) <-> (~P1 (/) = (/) /\ {(/)} e. 1c)))
15	4, 14	spcev 2946	. . 3 |- ((~P1 (/) = (/) /\ {(/)} e. 1c) -> E.a(~P1 a e. 0c /\ ~Pa e. 1c))
16	3, 5, 15	mp2an 653	. 2 |- E.a(~P1 a e. 0c /\ ~Pa e. 1c)
17		df-sfin 4446	. 2 |- ( Sfin (0c, 1c) <-> (0c e. Nn /\ 1c e. Nn /\ E.a(~P1 a e. 0c /\ ~Pa e. 1c)))
18	1, 2, 16, 17	mpbir3an 1134	1 |- Sfin (0c, 1c)

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  (/)c0 3550  ~Pcpw 3722  {csn 3737  1_cc1c 4134  ~P₁ cpw1 4135   Nn cnnc 4373  0_cc0c 4374   S_fin wsfin 4438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193  df-0c 4377  df-addc 4378  df-nnc 4379  df-sfin 4446

This theorem is referenced by:  sfintfin  4532