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Theorem 0fin 4423

Description: The empty set is finite. (Contributed by SF, 19-Jan-2015.)

Assertion

Ref	Expression
0fin	⊢ ∅ ∈ Fin

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Proof of Theorem 0fin

Step	Hyp	Ref	Expression
1		peano1 4402	. . 3 ⊢ 0c ∈ Nn
2		eqid 2353	. . . 4 ⊢ ∅ = ∅
3		el0c 4421	. . . 4 ⊢ (∅ ∈ 0c ↔ ∅ = ∅)
4	2, 3	mpbir 200	. . 3 ⊢ ∅ ∈ 0c
5		eleq2 2414	. . . 4 ⊢ (n = 0c → (∅ ∈ n ↔ ∅ ∈ 0c))
6	5	rspcev 2955	. . 3 ⊢ ((0c ∈ Nn ∧ ∅ ∈ 0c) → ∃n ∈ Nn ∅ ∈ n)
7	1, 4, 6	mp2an 653	. 2 ⊢ ∃n ∈ Nn ∅ ∈ n
8		elfin 4420	. 2 ⊢ (∅ ∈ Fin ↔ ∃n ∈ Nn ∅ ∈ n)
9	7, 8	mpbir 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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