Theorem elgch 9444

Description: Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
elgch	|- ( A e. V -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem elgch

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-gch 9443	. . . 4 |- GCH = ( Fin u. { y | A. x -. ( y ~< x /\ x ~< ~P y ) } )
2	1	eleq2i 2693	. . 3 |- ( A e. GCH <-> A e. ( Fin u. { y | A. x -. ( y ~< x /\ x ~< ~P y ) } ) )
3		elun 3753	. . 3 |- ( A e. ( Fin u. { y | A. x -. ( y ~< x /\ x ~< ~P y ) } ) <-> ( A e. Fin \/ A e. { y | A. x -. ( y ~< x /\ x ~< ~P y ) } ) )
4	2, 3	bitri 264	. 2 |- ( A e. GCH <-> ( A e. Fin \/ A e. { y | A. x -. ( y ~< x /\ x ~< ~P y ) } ) )
5		breq1 4656	. . . . . . 7 |- ( y = A -> ( y ~< x <-> A ~< x ) )
6		pweq 4161	. . . . . . . 8 |- ( y = A -> ~P y = ~P A )
7	6	breq2d 4665	. . . . . . 7 |- ( y = A -> ( x ~< ~P y <-> x ~< ~P A ) )
8	5, 7	anbi12d 747	. . . . . 6 |- ( y = A -> ( ( y ~< x /\ x ~< ~P y ) <-> ( A ~< x /\ x ~< ~P A ) ) )
9	8	notbid 308	. . . . 5 |- ( y = A -> ( -. ( y ~< x /\ x ~< ~P y ) <-> -. ( A ~< x /\ x ~< ~P A ) ) )
10	9	albidv 1849	. . . 4 |- ( y = A -> ( A. x -. ( y ~< x /\ x ~< ~P y ) <-> A. x -. ( A ~< x /\ x ~< ~P A ) ) )
11	10	elabg 3351	. . 3 |- ( A e. V -> ( A e. { y | A. x -. ( y ~< x /\ x ~< ~P y ) } <-> A. x -. ( A ~< x /\ x ~< ~P A ) ) )
12	11	orbi2d 738	. 2 |- ( A e. V -> ( ( A e. Fin \/ A e. { y | A. x -. ( y ~< x /\ x ~< ~P y ) } ) <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
13	4, 12	syl5bb 272	1 |- ( A e. V -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   u. cun 3572   ~Pcpw 4158   class class class wbr 4653   ~< csdm 7954   Fincfn 7955  GCHcgch 9442

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

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-gch 9443

This theorem is referenced by:  gchi  9446  engch  9450  hargch  9495  alephgch  9496