Theorem isfi 8915

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 8890	. . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}
2	1	eleq2i 2829	. 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥})
3		relen 8891	. . . . 5 ⊢ Rel ≈
4	3	brrelex1i 5680	. . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V)
5	4	rexlimivw 3135	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V)
6		breq1 5089	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥))
7	6	rexbidv 3162	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥))
8	5, 7	elab3 3630	. 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 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3430   class class class wbr 5086  ωcom 7810   ≈ cen 8883  Fincfn 8886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-en 8887  df-fin 8890

This theorem is referenced by:  0fi  8982  snfi  8983  findcard  9091  findcard2  9092  nnfi  9095  ssnnfi  9097  unfi  9098  ssfiALT  9101  enfii  9113  enfiALT  9115  php3  9136  onfin  9142  ominf  9167  isinf  9168  dif1ennnALT  9180  findcard3  9186  nnsdomg  9202  isfiniteg  9203  prfi  9227  fiint  9230  finnum  9863  ficardom  9876  dif1card  9923  infpwfien  9975  ficard  10478  hashkf  14285  finminlem  36516  domalom  37734