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Theorem nnfi 6695
Description: Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Assertion
Ref Expression
nnfi (𝐴 ∈ ω → 𝐴 ∈ Fin)

Proof of Theorem nnfi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 enrefg 6588 . . 3 (𝐴 ∈ ω → 𝐴𝐴)
2 breq2 3879 . . . 4 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
32rspcev 2744 . . 3 ((𝐴 ∈ ω ∧ 𝐴𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
41, 3mpdan 415 . 2 (𝐴 ∈ ω → ∃𝑥 ∈ ω 𝐴𝑥)
5 isfi 6585 . 2 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
64, 5sylibr 133 1 (𝐴 ∈ ω → 𝐴 ∈ Fin)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1448  wrex 2376   class class class wbr 3875  ωcom 4442  cen 6562  Fincfn 6564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-en 6565  df-fin 6567
This theorem is referenced by:  dif1en  6702  0fin  6707  findcard2  6712  findcard2s  6713  diffisn  6716  en1eqsn  6764  fzfig  10044  hashennnuni  10366  hashennn  10367  hashun  10392  hashp1i  10397  pwf1oexmid  12780
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