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Theorem ncfinsn 4476
 Description: If the universe is finite, then the cardinality of a singleton is 1c. (Contributed by SF, 30-Jan-2015.)
Assertion
Ref Expression
ncfinsn ((V Fin A V) → Ncfin {A} = 1c)

Proof of Theorem ncfinsn
StepHypRef Expression
1 snex 4111 . . . . 5 {A} V
2 ncfinprop 4474 . . . . 5 ((V Fin {A} V) → ( Ncfin {A} Nn {A} Ncfin {A}))
31, 2mpan2 652 . . . 4 (V Fin → ( Ncfin {A} Nn {A} Ncfin {A}))
43adantr 451 . . 3 ((V Fin A V) → ( Ncfin {A} Nn {A} Ncfin {A}))
54simpld 445 . 2 ((V Fin A V) → Ncfin {A} Nn )
6 1cnnc 4408 . . 3 1c Nn
76a1i 10 . 2 ((V Fin A V) → 1c Nn )
84simprd 449 . 2 ((V Fin A V) → {A} Ncfin {A})
9 snel1cg 4141 . . 3 (A V → {A} 1c)
109adantl 452 . 2 ((V Fin A V) → {A} 1c)
11 nnceleq 4430 . 2 ((( Ncfin {A} Nn 1c Nn ) ({A} Ncfin {A} {A} 1c)) → Ncfin {A} = 1c)
125, 7, 8, 10, 11syl22anc 1183 1 ((V Fin A V) → Ncfin {A} = 1c)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737  1cc1c 4134   Nn cnnc 4373   Fin cfin 4376   Ncfin cncfin 4434 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 This theorem is referenced by:  vfinncsp  4554
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