Theorem tncveqnc1fin 4544

Description: If the universe is finite, then the T-raising of the size of the universe is equal to the size of 1_c. Theorem X.1.55 of [Rosser] p. 534. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
tncveqnc1fin	⊢ (V ∈ Fin → Tfin Ncfin V = Ncfin 1c)

Proof of Theorem tncveqnc1fin

Step	Hyp	Ref	Expression
1		vvex 4109	. . 3 ⊢ V ∈ V
2		ncfintfin 4495	. . 3 ⊢ ((V ∈ Fin ∧ V ∈ V) → Tfin Ncfin V = Ncfin ℘1V)
3	1, 2	mpan2 652	. 2 ⊢ (V ∈ Fin → Tfin Ncfin V = Ncfin ℘1V)
4		df1c2 4168	. . 3 ⊢ 1c = ℘1V
5		ncfineq 4473	. . 3 ⊢ (1c = ℘1V → Ncfin 1c = Ncfin ℘1V)
6	4, 5	ax-mp 8	. 2 ⊢ Ncfin 1c = Ncfin ℘1V
7	3, 6	syl6eqr 2403	1 ⊢ (V ∈ Fin → Tfin Ncfin V = Ncfin 1c)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859  1_cc1c 4134  ℘₁cpw1 4135   Fin cfin 4376   Nc_fin cncfin 4434   T_fin ctfin 4435

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-tfin 4443

This theorem is referenced by:  t1csfin1c  4545  vfintle  4546  vfinspss  4551  vfinspclt  4552