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Theorem 1cvsfin 4542
 Description: If the universe is finite, then Ncfin 1c is the base two log of Ncfin V. Theorem X.1.54 of [Rosser] p. 534. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
1cvsfin (V FinSfin ( Ncfin 1c, Ncfin V))

Proof of Theorem 1cvsfin
StepHypRef Expression
1 1cex 4142 . . . 4 1c V
2 ncfinprop 4474 . . . . 5 ((V Fin 1c V) → ( Ncfin 1c Nn 1c Ncfin 1c))
32simpld 445 . . . 4 ((V Fin 1c V) → Ncfin 1c Nn )
41, 3mpan2 652 . . 3 (V FinNcfin 1c Nn )
5 vvex 4109 . . . 4 V V
6 ncfinprop 4474 . . . . 5 ((V Fin V V) → ( Ncfin V Nn V Ncfin V))
76simpld 445 . . . 4 ((V Fin V V) → Ncfin V Nn )
85, 7mpan2 652 . . 3 (V FinNcfin V Nn )
91, 2mpan2 652 . . . . 5 (V Fin → ( Ncfin 1c Nn 1c Ncfin 1c))
109simprd 449 . . . 4 (V Fin → 1c Ncfin 1c)
115, 6mpan2 652 . . . . 5 (V Fin → ( Ncfin V Nn V Ncfin V))
1211simprd 449 . . . 4 (V Fin → V Ncfin V)
13 pw1eq 4143 . . . . . . . 8 (a = V → 1a = 1V)
14 df1c2 4168 . . . . . . . 8 1c = 1V
1513, 14syl6eqr 2403 . . . . . . 7 (a = V → 1a = 1c)
1615eleq1d 2419 . . . . . 6 (a = V → (1a Ncfin 1c ↔ 1c Ncfin 1c))
17 pweq 3725 . . . . . . . 8 (a = V → a = V)
18 pwv 3886 . . . . . . . 8 V = V
1917, 18syl6eq 2401 . . . . . . 7 (a = V → a = V)
2019eleq1d 2419 . . . . . 6 (a = V → (a Ncfin V ↔ V Ncfin V))
2116, 20anbi12d 691 . . . . 5 (a = V → ((1a Ncfin 1c a Ncfin V) ↔ (1c Ncfin 1c V Ncfin V)))
225, 21spcev 2946 . . . 4 ((1c Ncfin 1c V Ncfin V) → a(1a Ncfin 1c a Ncfin V))
2310, 12, 22syl2anc 642 . . 3 (V Fina(1a Ncfin 1c a Ncfin V))
244, 8, 233jca 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)))
2624, 25sylibr 203 1 (V FinSfin ( Ncfin 1c, Ncfin V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ℘cpw 3722  1cc1c 4134  ℘1cpw1 4135   Nn cnnc 4373   Fin cfin 4376   Ncfin cncfin 4434   Sfin 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
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