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Theorem 0fin 4423
 Description: The empty set is finite. (Contributed by SF, 19-Jan-2015.)
Assertion
Ref Expression
0fin Fin

Proof of Theorem 0fin
StepHypRef Expression
1 peano1 4402 . . 3 0c Nn
2 eqid 2353 . . . 4 =
3 el0c 4421 . . . 4 ( 0c = )
42, 3mpbir 200 . . 3 0c
5 eleq2 2414 . . . 4 (n = 0c → ( n 0c))
65rspcev 2955 . . 3 ((0c Nn 0c) → n Nn n)
71, 4, 6mp2an 653 . 2 n Nn n
8 elfin 4420 . 2 ( Finn Nn n)
97, 8mpbir 200 1 Fin
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∅c0 3550   Nn cnnc 4373  0cc0c 4374   Fin cfin 4376 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892  df-int 3927  df-0c 4377  df-nnc 4379  df-fin 4380 This theorem is referenced by:  snfi  4431  ssfin  4470  nchoicelem18  6306
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