Theorem sfin01 4529

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 4403	. 2 |- 0c e. Nn
2		1cnnc 4409	. 2 |- 1c e. Nn
3		pw10 4162	. . 3 |- ~P1 (/) = (/)
4		0ex 4111	. . . 4 |- (/) e. _V
5	4	snel1c 4141	. . 3 |- {(/)} e. 1c
6		el0c 4422	. . . . . 6 |- (~P1 a e. 0c <-> ~P1 a = (/))
7		pw1eq 4144	. . . . . . 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 3726	. . . . . . 7 |- (a = (/) -> ~Pa = ~P(/))
11		pw0 4161	. . . . . . 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 2947	. . 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 4447	. 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 3551  ~Pcpw 3723  {csn 3738  1_cc1c 4135  ~P₁ cpw1 4136   Nn cnnc 4374  0_cc0c 4375   S_fin wsfin 4439

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-ins2 4085  ax-ins3 4086  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-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194  df-0c 4378  df-addc 4379  df-nnc 4380  df-sfin 4447

This theorem is referenced by:  sfintfin  4533