Theorem 1cvsfin 4542

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 4142	. . . 4 |- 1c e. _V
2		ncfinprop 4474	. . . . 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 4109	. . . 4 |- _V e. _V
6		ncfinprop 4474	. . . . 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 4143	. . . . . . . 8 |- (a = _V -> ~P1 a = ~P1 _V)
14		df1c2 4168	. . . . . . . 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 3725	. . . . . . . 8 |- (a = _V -> ~Pa = ~P_V)
18		pwv 3886	. . . . . . . 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 2946	. . . 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 4446	. 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 2859  ~Pcpw 3722  1_cc1c 4134  ~P₁ cpw1 4135   Nn cnnc 4373   Fin cfin 4376   Nc_fin cncfin 4434   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-14 1714  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-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  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-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-ncfin 4442  df-sfin 4446

This theorem is referenced by:  1cspfin  4543  t1csfin1c  4545  vfinspss  4551