Theorem 1cvsfin 4543

Description: If the universe is finite, then Nc_fin 1_c is the base two log of Nc_fin _V. Theorem X.1.54 of [Rosser] p. 534. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
1cvsfin	|- (_V e. Fin -> Sfin ( Ncfin 1c, Ncfin _V))

Proof of Theorem 1cvsfin

Step	Hyp	Ref	Expression
1		1cex 4143	. . . 4 |- 1c e. _V
2		ncfinprop 4475	. . . . 5 |- ((_V e. Fin /\ 1c e. _V) -> ( Ncfin 1c e. Nn /\ 1c e. Ncfin 1c))
3	2	simpld 445	. . . 4 |- ((_V e. Fin /\ 1c e. _V) -> Ncfin 1c e. Nn )
4	1, 3	mpan2 652	. . 3 |- (_V e. Fin -> Ncfin 1c e. Nn )
5		vvex 4110	. . . 4 |- _V e. _V
6		ncfinprop 4475	. . . . 5 |- ((_V e. Fin /\ _V e. _V) -> ( Ncfin _V e. Nn /\ _V e. Ncfin _V))
7	6	simpld 445	. . . 4 |- ((_V e. Fin /\ _V e. _V) -> Ncfin _V e. Nn )
8	5, 7	mpan2 652	. . 3 |- (_V e. Fin -> Ncfin _V e. Nn )
9	1, 2	mpan2 652	. . . . 5 |- (_V e. Fin -> ( Ncfin 1c e. Nn /\ 1c e. Ncfin 1c))
10	9	simprd 449	. . . 4 |- (_V e. Fin -> 1c e. Ncfin 1c)
11	5, 6	mpan2 652	. . . . 5 |- (_V e. Fin -> ( Ncfin _V e. Nn /\ _V e. Ncfin _V))
12	11	simprd 449	. . . 4 |- (_V e. Fin -> _V e. Ncfin _V)
13		pw1eq 4144	. . . . . . . 8 |- (a = _V -> ~P1 a = ~P1 _V)
14		df1c2 4169	. . . . . . . 8 |- 1c = ~P1 _V
15	13, 14	syl6eqr 2403	. . . . . . 7 |- (a = _V -> ~P1 a = 1c)
16	15	eleq1d 2419	. . . . . 6 |- (a = _V -> (~P1 a e. Ncfin 1c <-> 1c e. Ncfin 1c))
17		pweq 3726	. . . . . . . 8 |- (a = _V -> ~Pa = ~P_V)
18		pwv 3887	. . . . . . . 8 |- ~P_V = _V
19	17, 18	syl6eq 2401	. . . . . . 7 |- (a = _V -> ~Pa = _V)
20	19	eleq1d 2419	. . . . . 6 |- (a = _V -> (~Pa e. Ncfin _V <-> _V e. Ncfin _V))
21	16, 20	anbi12d 691	. . . . 5 |- (a = _V -> ((~P1 a e. Ncfin 1c /\ ~Pa e. Ncfin _V) <-> (1c e. Ncfin 1c /\ _V e. Ncfin _V)))
22	5, 21	spcev 2947	. . . 4 |- ((1c e. Ncfin 1c /\ _V e. Ncfin _V) -> E.a(~P1 a e. Ncfin 1c /\ ~Pa e. Ncfin _V))
23	10, 12, 22	syl2anc 642	. . 3 |- (_V e. Fin -> E.a(~P1 a e. Ncfin 1c /\ ~Pa e. Ncfin _V))
24	4, 8, 23	3jca 1132	. 2 |- (_V e. Fin -> ( Ncfin 1c e. Nn /\ Ncfin _V e. Nn /\ E.a(~P1 a e. Ncfin 1c /\ ~Pa e. Ncfin _V)))
25		df-sfin 4447	. 2 |- ( Sfin ( Ncfin 1c, Ncfin _V) <-> ( Ncfin 1c e. Nn /\ Ncfin _V e. Nn /\ E.a(~P1 a e. Ncfin 1c /\ ~Pa e. Ncfin _V)))
26	24, 25	sylibr 203	1 |- (_V e. Fin -> Sfin ( Ncfin 1c, Ncfin _V))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860  ~Pcpw 3723  1_cc1c 4135  ~P₁ cpw1 4136   Nn cnnc 4374   Fin cfin 4377   Nc_fin cncfin 4435   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-14 1714  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-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-ncfin 4443  df-sfin 4447

This theorem is referenced by:  1cspfin  4544  t1csfin1c  4546  vfinspss  4552