Theorem elfin 4421

Description: Membership in the set of finite sets. (Contributed by SF, 19-Jan-2015.)

Assertion

Ref	Expression
elfin	⊢ (A ∈ Fin ↔ ∃x ∈ Nn A ∈ x)

Distinct variable group:   x,A

Proof of Theorem elfin

Step	Hyp	Ref	Expression
1		df-fin 4381	. . 3 ⊢ Fin = ∪ Nn
2	1	eleq2i 2417	. 2 ⊢ (A ∈ Fin ↔ A ∈ ∪ Nn )
3		eluni2 3896	. 2 ⊢ (A ∈ ∪ Nn ↔ ∃x ∈ Nn A ∈ x)
4	2, 3	bitri 240	1 ⊢ (A ∈ Fin ↔ ∃x ∈ Nn A ∈ x)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∃wrex 2616  ∪cuni 3892   Nn cnnc 4374   Fin cfin 4377

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

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 2479  df-rex 2621  df-v 2862  df-uni 3893  df-fin 4381

This theorem is referenced by:  0fin  4424  snfi  4432  ssfin  4471  vfinnc  4472  sfinltfin  4536  ncssfin  6152  pw1fin  6170  nntccl  6171  finnc  6244  ncfin  6248