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Theorem vfinncvntsp 4549
 Description: If the universe is finite, then its size is not a T raising of an element of Spfin. Corollary of theorem X.1.58 of [Rosser] p. 534. (Contributed by SF, 27-Jan-2015.)
Assertion
Ref Expression
vfinncvntsp (V Fin → ¬ Ncfin V {a x Spfin a = Tfin x})
Distinct variable group:   x,a

Proof of Theorem vfinncvntsp
StepHypRef Expression
1 vfinspnn 4541 . . . . . . . 8 (V FinSpfin ( Nn {}))
2 difss 3393 . . . . . . . 8 ( Nn {}) Nn
31, 2syl6ss 3284 . . . . . . 7 (V FinSpfin Nn )
43sselda 3273 . . . . . 6 ((V Fin x Spfin ) → x Nn )
5 vfinncvntnn 4548 . . . . . 6 ((V Fin x Nn ) → Tfin xNcfin V)
64, 5syldan 456 . . . . 5 ((V Fin x Spfin ) → Tfin xNcfin V)
76necomd 2599 . . . 4 ((V Fin x Spfin ) → Ncfin V ≠ Tfin x)
8 df-ne 2518 . . . 4 ( Ncfin V ≠ Tfin x ↔ ¬ Ncfin V = Tfin x)
97, 8sylib 188 . . 3 ((V Fin x Spfin ) → ¬ Ncfin V = Tfin x)
109nrexdv 2717 . 2 (V Fin → ¬ x Spfin Ncfin V = Tfin x)
11 ncfinex 4472 . . 3 Ncfin V V
12 eqeq1 2359 . . . 4 (a = Ncfin V → (a = Tfin xNcfin V = Tfin x))
1312rexbidv 2635 . . 3 (a = Ncfin V → (x Spfin a = Tfin xx Spfin Ncfin V = Tfin x))
1411, 13elab 2985 . 2 ( Ncfin V {a x Spfin a = Tfin x} ↔ x Spfin Ncfin V = Tfin x)
1510, 14sylnibr 296 1 (V Fin → ¬ Ncfin V {a x Spfin a = Tfin x})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ∖ cdif 3206  ∅c0 3550  {csn 3737   Nn cnnc 4373   Fin cfin 4376   Ncfin cncfin 4434   Tfin ctfin 4435   Spfin cspfin 4439 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-3or 935  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-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-sfin 4446  df-spfin 4447 This theorem is referenced by:  vfinncsp  4554
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