Theorem isfin3 9118

Description: Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
isfin3	|- ( A e. FinIII <-> ~P A e. FinIV )

Proof of Theorem isfin3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fin3 9110	. . 3 |- FinIII = { x | ~P x e. FinIV }
2	1	eleq2i 2693	. 2 |- ( A e. FinIII <-> A e. { x | ~P x e. FinIV } )
3		pwexr 6974	. . 3 |- ( ~P A e. FinIV -> A e. _V )
4		pweq 4161	. . . 4 |- ( x = A -> ~P x = ~P A )
5	4	eleq1d 2686	. . 3 |- ( x = A -> ( ~P x e. FinIV <-> ~P A e. FinIV ) )
6	3, 5	elab3 3358	. 2 |- ( A e. { x | ~P x e. FinIV } <-> ~P A e. FinIV )
7	2, 6	bitri 264	1 |- ( A e. FinIII <-> ~P A e. FinIV )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   ~Pcpw 4158  Fin^IVcfin4 9102  Fin^IIIcfin3 9103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-fin3 9110

This theorem is referenced by:  fin23lem41  9174  isfin32i  9187  fin34  9212