Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin3 Structured version   Unicode version

Theorem isfin3 8674
 Description: Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin3 FinIII FinIV

Proof of Theorem isfin3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-fin3 8666 . . 3 FinIII FinIV
21eleq2i 2519 . 2 FinIII FinIV
3 elex 3102 . . . 4 FinIV
4 pwexb 6592 . . . 4
53, 4sylibr 212 . . 3 FinIV
6 pweq 3996 . . . 4
76eleq1d 2510 . . 3 FinIV FinIV
85, 7elab3 3237 . 2 FinIV FinIV
92, 8bitri 249 1 FinIII FinIV
 Colors of variables: wff setvar class Syntax hints:   wb 184   wceq 1381   wcel 1802  cab 2426  cvv 3093  cpw 3993  FinIVcfin4 8658  FinIIIcfin3 8659 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rex 2797  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-pw 3995  df-sn 4011  df-pr 4013  df-uni 4231  df-fin3 8666 This theorem is referenced by:  fin23lem41  8730  isfin32i  8743  fin34  8768
 Copyright terms: Public domain W3C validator