Theorem t1csfin1c 4546

Description: If the universe is finite, then the T-raising of the size of 1_c is smaller than the size itself. Corollary of theorem X.1.56 of [Rosser] p. 534. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
t1csfin1c	⊢ (V ∈ Fin → Sfin ( Tfin Ncfin 1c, Ncfin 1c))

Proof of Theorem t1csfin1c

Step	Hyp	Ref	Expression
1		1cvsfin 4543	. . 3 ⊢ (V ∈ Fin → Sfin ( Ncfin 1c, Ncfin V))
2		sfintfin 4533	. . 3 ⊢ ( Sfin ( Ncfin 1c, Ncfin V) → Sfin ( Tfin Ncfin 1c, Tfin Ncfin V))
3	1, 2	syl 15	. 2 ⊢ (V ∈ Fin → Sfin ( Tfin Ncfin 1c, Tfin Ncfin V))
4		tncveqnc1fin 4545	. . 3 ⊢ (V ∈ Fin → Tfin Ncfin V = Ncfin 1c)
5		sfineq2 4528	. . 3 ⊢ ( Tfin Ncfin V = Ncfin 1c → ( Sfin ( Tfin Ncfin 1c, Tfin Ncfin V) ↔ Sfin ( Tfin Ncfin 1c, Ncfin 1c)))
6	4, 5	syl 15	. 2 ⊢ (V ∈ Fin → ( Sfin ( Tfin Ncfin 1c, Tfin Ncfin V) ↔ Sfin ( Tfin Ncfin 1c, Ncfin 1c)))
7	3, 6	mpbid 201	1 ⊢ (V ∈ Fin → Sfin ( Tfin Ncfin 1c, Ncfin 1c))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860  1_cc1c 4135   Fin cfin 4377   Nc_fin cncfin 4435   T_fin ctfin 4436   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-13 1712  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-tfin 4444  df-sfin 4447

This theorem is referenced by:  vfinspsslem1  4551