Theorem isfi 6815

Description: Express "𝐴 is finite". Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)

Assertion

Ref	Expression
isfi	⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem isfi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fin 6797	. . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}
2	1	eleq2i 2260	. 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥})
3		relen 6798	. . . . 5 ⊢ Rel ≈
4	3	brrelex1i 4702	. . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V)
5	4	rexlimivw 2607	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V)
6		breq1 4032	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥))
7	6	rexbidv 2495	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥))
8	5, 7	elab3 2912	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
9	2, 8	bitri 184	1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473  Vcvv 2760   class class class wbr 4029  ωcom 4622   ≈ cen 6792  Fincfn 6794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-en 6795  df-fin 6797

This theorem is referenced by:  snfig  6868  fict  6924  fidceq  6925  nnfi  6928  enfi  6929  ssfilem  6931  dif1enen  6936  php5fin  6938  fisbth  6939  fin0  6941  fin0or  6942  diffitest  6943  findcard  6944  findcard2  6945  findcard2s  6946  diffisn  6949  infnfi  6951  fientri3  6971  unsnfi  6975  unsnfidcex  6976  unsnfidcel  6977  fiintim  6985  fidcenumlemim  7011  finnum  7243  hashcl  10852  hashen  10855  fihashdom  10874  hashun  10876  zfz1iso  10912