Theorem isfi 7013

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 6991	. . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}
2	1	eleq2i 2301	. 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥})
3		relen 6992	. . . . 5 ⊢ Rel ≈
4	3	brrelex1i 4798	. . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V)
5	4	rexlimivw 2658	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V)
6		breq1 4117	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥))
7	6	rexbidv 2545	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥))
8	5, 7	elab3 2972	. 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 1398   ∈ wcel 2205  {cab 2220  ∃wrex 2523  Vcvv 2815   class class class wbr 4114  ωcom 4717   ≈ cen 6986  Fincfn 6988

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-en 6989  df-fin 6991

This theorem is referenced by:  snfig  7069  fict  7136  fidceq  7137  nnfi  7140  enfi  7141  ssfilem  7143  ssfilemd  7145  dif1enen  7150  php5fin  7152  fisbth  7153  fin0  7155  fin0or  7156  diffitest  7157  findcard  7158  findcard2  7159  findcard2s  7160  diffisn  7163  infnfi  7165  fidcen  7169  fientri3  7188  unsnfi  7192  unsnfidcex  7193  unsnfidcel  7194  fiintim  7204  fidcenumlemim  7235  finnum  7492  ficardon  7498  hashcl  11169  hashen  11172  fihashdom  11192  hashun  11194  zfz1iso  11238