Theorem isfi 8893

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 8868	. . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}
2	1	eleq2i 2823	. 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥})
3		relen 8869	. . . . 5 ⊢ Rel ≈
4	3	brrelex1i 5667	. . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V)
5	4	rexlimivw 3129	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V)
6		breq1 5089	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥))
7	6	rexbidv 3156	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥))
8	5, 7	elab3 3637	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
9	2, 8	bitri 275	1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056  Vcvv 3436   class class class wbr 5086  ωcom 7791   ≈ cen 8861  Fincfn 8864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-en 8865  df-fin 8868

This theorem is referenced by:  0fi  8959  snfi  8960  findcard  9068  findcard2  9069  nnfi  9072  ssnnfi  9074  unfi  9075  ssfiALT  9078  enfii  9090  enfiALT  9092  php3  9113  onfin  9119  ominf  9143  isinf  9144  dif1ennnALT  9156  findcard3  9162  nnsdomg  9178  isfiniteg  9179  prfi  9203  fiint  9206  finnum  9836  ficardom  9849  dif1card  9896  infpwfien  9948  ficard  10451  hashkf  14234  finminlem  36352  domalom  37438