Theorem sbthlem10 8079

Description: Lemma for sbth 8080. (Contributed by NM, 28-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	|- A e. _V
sbthlem.2	|- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
sbthlem.3	|- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
sbthlem.4	|- B e. _V

Assertion

Ref	Expression
sbthlem10	|- ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B )

Distinct variable groups:   x, A   x, B   x, D   x, f, g   x, H   f, g, A   B, f, g

Allowed substitution hints:   D( f, g)   H( f, g)

Proof of Theorem sbthlem10

Step	Hyp	Ref	Expression
1		sbthlem.4	. . . . 5 |- B e. _V
2	1	brdom 7967	. . . 4 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		sbthlem.1	. . . . 5 |- A e. _V
4	3	brdom 7967	. . . 4 |- ( B ~<_ A <-> E. g g : B -1-1-> A )
5	2, 4	anbi12i 733	. . 3 |- ( ( A ~<_ B /\ B ~<_ A ) <-> ( E. f f : A -1-1-> B /\ E. g g : B -1-1-> A ) )
6		eeanv 2182	. . 3 |- ( E. f E. g ( f : A -1-1-> B /\ g : B -1-1-> A ) <-> ( E. f f : A -1-1-> B /\ E. g g : B -1-1-> A ) )
7	5, 6	bitr4i 267	. 2 |- ( ( A ~<_ B /\ B ~<_ A ) <-> E. f E. g ( f : A -1-1-> B /\ g : B -1-1-> A ) )
8		sbthlem.3	. . . . 5 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
9		vex 3203	. . . . . . 7 |- f e. _V
10	9	resex 5443	. . . . . 6 |- ( f |` U. D ) e. _V
11		vex 3203	. . . . . . . 8 |- g e. _V
12	11	cnvex 7113	. . . . . . 7 |- `' g e. _V
13	12	resex 5443	. . . . . 6 |- ( `' g |` ( A \ U. D ) ) e. _V
14	10, 13	unex 6956	. . . . 5 |- ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) e. _V
15	8, 14	eqeltri 2697	. . . 4 |- H e. _V
16		sbthlem.2	. . . . 5 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
17	3, 16, 8	sbthlem9 8078	. . . 4 |- ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )
18		f1oen3g 7971	. . . 4 |- ( ( H e. _V /\ H : A -1-1-onto-> B ) -> A ~~ B )
19	15, 17, 18	sylancr 695	. . 3 |- ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> A ~~ B )
20	19	exlimivv 1860	. 2 |- ( E. f E. g ( f : A -1-1-> B /\ g : B -1-1-> A ) -> A ~~ B )
21	7, 20	sylbi 207	1 |- ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   |` cres 5116   "cima 5117   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ~~ cen 7952   ~<_ cdom 7953

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-pow 4843  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-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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-en 7956  df-dom 7957

This theorem is referenced by:  sbth  8080