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Theorem elfin 4420
 Description: Membership in the set of finite sets. (Contributed by SF, 19-Jan-2015.)
Assertion
Ref Expression
elfin (A Finx Nn A x)
Distinct variable group:   x,A

Proof of Theorem elfin
StepHypRef Expression
1 df-fin 4380 . . 3 Fin = Nn
21eleq2i 2417 . 2 (A FinA Nn )
3 eluni2 3895 . 2 (A Nnx Nn A x)
42, 3bitri 240 1 (A Finx Nn A x)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∃wrex 2615  ∪cuni 3891   Nn cnnc 4373   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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rex 2620  df-v 2861  df-uni 3892  df-fin 4380 This theorem is referenced by:  0fin  4423  snfi  4431  ssfin  4470  vfinnc  4471  sfinltfin  4535  ncssfin  6151  pw1fin  6169  nntccl  6170  finnc  6243  ncfin  6247
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