Theorem isfin6 9122

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

Assertion

Ref	Expression
isfin6	|- ( A e. FinVI <-> ( A ~< 2o \/ A ~< ( A X. A ) ) )

Proof of Theorem isfin6

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fin6 9112	. . 3 |- FinVI = { x | ( x ~< 2o \/ x ~< ( x X. x ) ) }
2	1	eleq2i 2693	. 2 |- ( A e. FinVI <-> A e. { x | ( x ~< 2o \/ x ~< ( x X. x ) ) } )
3		relsdom 7962	. . . . 5 |- Rel ~<
4	3	brrelexi 5158	. . . 4 |- ( A ~< 2o -> A e. _V )
5	3	brrelexi 5158	. . . 4 |- ( A ~< ( A X. A ) -> A e. _V )
6	4, 5	jaoi 394	. . 3 |- ( ( A ~< 2o \/ A ~< ( A X. A ) ) -> A e. _V )
7		breq1 4656	. . . 4 |- ( x = A -> ( x ~< 2o <-> A ~< 2o ) )
8		id 22	. . . . 5 |- ( x = A -> x = A )
9	8	sqxpeqd 5141	. . . . 5 |- ( x = A -> ( x X. x ) = ( A X. A ) )
10	8, 9	breq12d 4666	. . . 4 |- ( x = A -> ( x ~< ( x X. x ) <-> A ~< ( A X. A ) ) )
11	7, 10	orbi12d 746	. . 3 |- ( x = A -> ( ( x ~< 2o \/ x ~< ( x X. x ) ) <-> ( A ~< 2o \/ A ~< ( A X. A ) ) ) )
12	6, 11	elab3 3358	. 2 |- ( A e. { x | ( x ~< 2o \/ x ~< ( x X. x ) ) } <-> ( A ~< 2o \/ A ~< ( A X. A ) ) )
13	2, 12	bitri 264	1 |- ( A e. FinVI <-> ( A ~< 2o \/ A ~< ( A X. A ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   class class class wbr 4653   X. cxp 5112   2oc2o 7554   ~< csdm 7954  Fin^VIcfin6 9105

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-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

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dom 7957  df-sdom 7958  df-fin6 9112

This theorem is referenced by:  fin56  9215  fin67  9217