Theorem sbthlemi10 6995

Description: Lemma for isbth 6996. (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 6776	. . . . 5 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		sbthlem.1	. . . . . 6 |- A e. _V
4	3	brdom 6776	. . . . 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 1944	. . . 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 2755	. . . . . . . . 9 |- f e. _V
10	9	resex 4966	. . . . . . . 8 |- ( f |` U. D ) e. _V
11		vex 2755	. . . . . . . . . 10 |- g e. _V
12	11	cnvex 5185	. . . . . . . . 9 |- `' g e. _V
13	12	resex 4966	. . . . . . . 8 |- ( `' g |` ( A \ U. D ) ) e. _V
14	10, 13	unex 4459	. . . . . . 7 |- ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) e. _V
15	8, 14	eqeltri 2262	. . . . . 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 6994	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )
18		f1oen3g 6780	. . . . . 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 1208	. . . 4 |- (EXMID -> ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> A ~~ B ) )
21	20	exlimdvv 1909	. . 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 980   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   _Vcvv 2752   \ cdif 3141   u. cun 3142   C_ wss 3144   U.cuni 3824   class class class wbr 4018  EXMIDwem 4212   `'ccnv 4643   |` cres 4646   "cima 4647   -1-1->wf1 5232   -1-1-onto->wf1o 5234   ~~ cen 6764   ~<_ cdom 6765

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-exmid 4213  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-en 6767  df-dom 6768

This theorem is referenced by:  isbth  6996