Theorem sbthlemi10 7236

Description: Lemma for isbth 7237. (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 6987	. . . . 5 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		sbthlem.1	. . . . . 6 |- A e. _V
4	3	brdom 6987	. . . . 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 1986	. . . 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 2816	. . . . . . . . 9 |- f e. _V
10	9	resex 5079	. . . . . . . 8 |- ( f |` U. D ) e. _V
11		vex 2816	. . . . . . . . . 10 |- g e. _V
12	11	cnvex 5301	. . . . . . . . 9 |- `' g e. _V
13	12	resex 5079	. . . . . . . 8 |- ( `' g |` ( A \ U. D ) ) e. _V
14	10, 13	unex 4562	. . . . . . 7 |- ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) e. _V
15	8, 14	eqeltri 2305	. . . . . 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 7235	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )
18		f1oen3g 6993	. . . . . 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 1233	. . . 4 |- (EXMID -> ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> A ~~ B ) )
21	20	exlimdvv 1947	. . 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 1005   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   _Vcvv 2813   \ cdif 3208   u. cun 3209   C_ wss 3211   U.cuni 3914   class class class wbr 4109  EXMIDwem 4307   `'ccnv 4748   |` cres 4751   "cima 4752   -1-1->wf1 5349   -1-1-onto->wf1o 5351   ~~ cen 6973   ~<_ cdom 6974

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-exmid 4308  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-en 6976  df-dom 6977

This theorem is referenced by:  isbth  7237