Theorem gchen1 9447

Description: If A <_ B < ~P A, and A is an infinite GCH-set, then A = B in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
gchen1	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B )

Proof of Theorem gchen1

Step	Hyp	Ref	Expression
1		simprl 794	. 2 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~<_ B )
2		gchi 9446	. . . . . . 7 |- ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )
3	2	3com23 1271	. . . . . 6 |- ( ( A e. GCH /\ B ~< ~P A /\ A ~< B ) -> A e. Fin )
4	3	3expia 1267	. . . . 5 |- ( ( A e. GCH /\ B ~< ~P A ) -> ( A ~< B -> A e. Fin ) )
5	4	con3dimp 457	. . . 4 |- ( ( ( A e. GCH /\ B ~< ~P A ) /\ -. A e. Fin ) -> -. A ~< B )
6	5	an32s 846	. . 3 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ B ~< ~P A ) -> -. A ~< B )
7	6	adantrl 752	. 2 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> -. A ~< B )
8		bren2 7986	. 2 |- ( A ~~ B <-> ( A ~<_ B /\ -. A ~< B ) )
9	1, 7, 8	sylanbrc 698	1 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   ~Pcpw 4158   class class class wbr 4653   ~~ cen 7952   ~<_ cdom 7953   ~< 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  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-eu 2474  df-mo 2475  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-f1o 5895  df-en 7956  df-dom 7957  df-sdom 7958  df-gch 9443

This theorem is referenced by:  gchor  9449  gchcda1  9478  gchcdaidm  9490  gchxpidm  9491  gchhar  9501