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 ∈ Fin → Sfin ( Ncfin 1c, Ncfin V))

Proof of Theorem 1cvsfin

Step	Hyp	Ref	Expression
1		1cex 4142	. . . 4 ⊢ 1c ∈ V
2		ncfinprop 4474	. . . . 5 ⊢ ((V ∈ Fin ∧ 1c ∈ V) → ( Ncfin 1c ∈ Nn ∧ 1c ∈ Ncfin 1c))
3	2	simpld 445	. . . 4 ⊢ ((V ∈ Fin ∧ 1c ∈ V) → Ncfin 1c ∈ Nn )
4	1, 3	mpan2 652	. . 3 ⊢ (V ∈ Fin → Ncfin 1c ∈ Nn )
5		vvex 4109	. . . 4 ⊢ V ∈ V
6		ncfinprop 4474	. . . . 5 ⊢ ((V ∈ Fin ∧ V ∈ V) → ( Ncfin V ∈ Nn ∧ V ∈ Ncfin V))
7	6	simpld 445	. . . 4 ⊢ ((V ∈ Fin ∧ V ∈ V) → Ncfin V ∈ Nn )
8	5, 7	mpan2 652	. . 3 ⊢ (V ∈ Fin → Ncfin V ∈ Nn )
9	1, 2	mpan2 652	. . . . 5 ⊢ (V ∈ Fin → ( Ncfin 1c ∈ Nn ∧ 1c ∈ Ncfin 1c))
10	9	simprd 449	. . . 4 ⊢ (V ∈ Fin → 1c ∈ Ncfin 1c)
11	5, 6	mpan2 652	. . . . 5 ⊢ (V ∈ Fin → ( Ncfin V ∈ Nn ∧ V ∈ Ncfin V))
12	11	simprd 449	. . . 4 ⊢ (V ∈ Fin → V ∈ Ncfin V)
13		pw1eq 4143	. . . . . . . 8 ⊢ (a = V → ℘1a = ℘1V)
14		df1c2 4168	. . . . . . . 8 ⊢ 1c = ℘1V
15	13, 14	syl6eqr 2403	. . . . . . 7 ⊢ (a = V → ℘1a = 1c)
16	15	eleq1d 2419	. . . . . 6 ⊢ (a = V → (℘1a ∈ Ncfin 1c ↔ 1c ∈ Ncfin 1c))
17		pweq 3725	. . . . . . . 8 ⊢ (a = V → ℘a = ℘V)
18		pwv 3886	. . . . . . . 8 ⊢ ℘V = V
19	17, 18	syl6eq 2401	. . . . . . 7 ⊢ (a = V → ℘a = V)
20	19	eleq1d 2419	. . . . . 6 ⊢ (a = V → (℘a ∈ Ncfin V ↔ V ∈ Ncfin V))
21	16, 20	anbi12d 691	. . . . 5 ⊢ (a = V → ((℘1a ∈ Ncfin 1c ∧ ℘a ∈ Ncfin V) ↔ (1c ∈ Ncfin 1c ∧ V ∈ Ncfin V)))
22	5, 21	spcev 2946	. . . 4 ⊢ ((1c ∈ Ncfin 1c ∧ V ∈ Ncfin V) → ∃a(℘1a ∈ Ncfin 1c ∧ ℘a ∈ Ncfin V))
23	10, 12, 22	syl2anc 642	. . 3 ⊢ (V ∈ Fin → ∃a(℘1a ∈ Ncfin 1c ∧ ℘a ∈ Ncfin V))
24	4, 8, 23	3jca 1132	. 2 ⊢ (V ∈ Fin → ( Ncfin 1c ∈ Nn ∧ Ncfin V ∈ Nn ∧ ∃a(℘1a ∈ Ncfin 1c ∧ ℘a ∈ Ncfin V)))
25		df-sfin 4446	. 2 ⊢ ( Sfin ( Ncfin 1c, Ncfin V) ↔ ( Ncfin 1c ∈ Nn ∧ Ncfin V ∈ Nn ∧ ∃a(℘1a ∈ Ncfin 1c ∧ ℘a ∈ Ncfin V)))
26	24, 25	sylibr 203	1 ⊢ (V ∈ Fin → Sfin ( Ncfin 1c, Ncfin V))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ℘cpw 3722  1_cc1c 4134  ℘₁cpw1 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