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Theorem snfi 4431
 Description: A singleton is finite. (Contributed by SF, 23-Feb-2015.)
Assertion
Ref Expression
snfi {A} Fin

Proof of Theorem snfi
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 1cnnc 4408 . . . 4 1c Nn
2 snel1cg 4141 . . . 4 (A V → {A} 1c)
3 eleq2 2414 . . . . 5 (x = 1c → ({A} x ↔ {A} 1c))
43rspcev 2955 . . . 4 ((1c Nn {A} 1c) → x Nn {A} x)
51, 2, 4sylancr 644 . . 3 (A V → x Nn {A} x)
6 elfin 4420 . . 3 ({A} Finx Nn {A} x)
75, 6sylibr 203 . 2 (A V → {A} Fin )
8 snprc 3788 . . 3 A V ↔ {A} = )
9 0fin 4423 . . . 4 Fin
10 eleq1 2413 . . . 4 ({A} = → ({A} Fin Fin ))
119, 10mpbiri 224 . . 3 ({A} = → {A} Fin )
128, 11sylbi 187 . 2 A V → {A} Fin )
137, 12pm2.61i 156 1 {A} Fin
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  ∅c0 3550  {csn 3737  1cc1c 4134   Nn cnnc 4373   Fin cfin 4376 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-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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-nul 3551  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-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380 This theorem is referenced by:  nchoicelem19  6307
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