Theorem sbthlemi10 7133

Description: Lemma for isbth 7134. (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
sbthlemi10	|- ( (EXMID /\ ( 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 sbthlemi10

Step	Hyp	Ref	Expression
1		sbthlem.4	. . . . . 6 |- B e. _V
2	1	brdom 6899	. . . . 5 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		sbthlem.1	. . . . . 6 |- A e. _V
4	3	brdom 6899	. . . . 5 |- ( B ~<_ A <-> E. g g : B -1-1-> A )
5	2, 4	anbi12i 460	. . . 4 |- ( ( A ~<_ B /\ B ~<_ A ) <-> ( E. f f : A -1-1-> B /\ E. g g : B -1-1-> A ) )
6		eeanv 1983	. . . 4 |- ( 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 187	. . 3 |- ( ( A ~<_ B /\ B ~<_ A ) <-> E. f E. g ( f : A -1-1-> B /\ g : B -1-1-> A ) )
8		sbthlem.3	. . . . . . 7 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
9		vex 2802	. . . . . . . . 9 |- f e. _V
10	9	resex 5046	. . . . . . . 8 |- ( f |` U. D ) e. _V
11		vex 2802	. . . . . . . . . 10 |- g e. _V
12	11	cnvex 5267	. . . . . . . . 9 |- `' g e. _V
13	12	resex 5046	. . . . . . . 8 |- ( `' g |` ( A \ U. D ) ) e. _V
14	10, 13	unex 4532	. . . . . . 7 |- ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) e. _V
15	8, 14	eqeltri 2302	. . . . . 6 |- H e. _V
16		sbthlem.2	. . . . . . 7 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
17	3, 16, 8	sbthlemi9 7132	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )
18		f1oen3g 6905	. . . . . 6 |- ( ( H e. _V /\ H : A -1-1-onto-> B ) -> A ~~ B )
19	15, 17, 18	sylancr 414	. . . . 5 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> A ~~ B )
20	19	3expib 1230	. . . 4 |- (EXMID -> ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> A ~~ B ) )
21	20	exlimdvv 1944	. . 3 |- (EXMID -> ( E. f E. g ( f : A -1-1-> B /\ g : B -1-1-> A ) -> A ~~ B ) )
22	7, 21	biimtrid 152	. 2 |- (EXMID -> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) )
23	22	imp 124	1 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   _Vcvv 2799   \ cdif 3194   u. cun 3195   C_ wss 3197   U.cuni 3888   class class class wbr 4083  EXMIDwem 4278   `'ccnv 4718   |` cres 4721   "cima 4722   -1-1->wf1 5315   -1-1-onto->wf1o 5317   ~~ cen 6885   ~<_ cdom 6886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-exmid 4279  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-en 6888  df-dom 6889

This theorem is referenced by:  isbth  7134