Theorem isbth 7134

Description: Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set A is smaller (has lower cardinality) than B and vice-versa, then A and B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7124 through sbthlemi10 7133; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7133. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 16391. (Contributed by NM, 8-Jun-1998.)

Assertion

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

Proof of Theorem isbth

Dummy variables x y z w f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 529	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~<_ B )
2		simprr 531	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> B ~<_ A )
3		reldom 6892	. . . . 5 |- Rel ~<_
4	3	brrelex1i 4762	. . . 4 |- ( B ~<_ A -> B e. _V )
5	2, 4	syl 14	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> B e. _V )
6		breq2 4087	. . . . . 6 |- ( w = B -> ( A ~<_ w <-> A ~<_ B ) )
7		breq1 4086	. . . . . 6 |- ( w = B -> ( w ~<_ A <-> B ~<_ A ) )
8	6, 7	anbi12d 473	. . . . 5 |- ( w = B -> ( ( A ~<_ w /\ w ~<_ A ) <-> ( A ~<_ B /\ B ~<_ A ) ) )
9		breq2 4087	. . . . 5 |- ( w = B -> ( A ~~ w <-> A ~~ B ) )
10	8, 9	imbi12d 234	. . . 4 |- ( w = B -> ( ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) <-> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) ) )
11	10	adantl 277	. . 3 |- ( ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) /\ w = B ) -> ( ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) <-> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) ) )
12	3	brrelex1i 4762	. . . . 5 |- ( A ~<_ B -> A e. _V )
13	1, 12	syl 14	. . . 4 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A e. _V )
14		breq1 4086	. . . . . . 7 |- ( z = A -> ( z ~<_ w <-> A ~<_ w ) )
15		breq2 4087	. . . . . . 7 |- ( z = A -> ( w ~<_ z <-> w ~<_ A ) )
16	14, 15	anbi12d 473	. . . . . 6 |- ( z = A -> ( ( z ~<_ w /\ w ~<_ z ) <-> ( A ~<_ w /\ w ~<_ A ) ) )
17		breq1 4086	. . . . . 6 |- ( z = A -> ( z ~~ w <-> A ~~ w ) )
18	16, 17	imbi12d 234	. . . . 5 |- ( z = A -> ( ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) <-> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) ) )
19	18	adantl 277	. . . 4 |- ( ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) /\ z = A ) -> ( ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) <-> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) ) )
20		vex 2802	. . . . . . 7 |- z e. _V
21		sseq1 3247	. . . . . . . . 9 |- ( y = x -> ( y C_ z <-> x C_ z ) )
22		imaeq2 5064	. . . . . . . . . . . 12 |- ( y = x -> ( f " y ) = ( f " x ) )
23	22	difeq2d 3322	. . . . . . . . . . 11 |- ( y = x -> ( w \ ( f " y ) ) = ( w \ ( f " x ) ) )
24	23	imaeq2d 5068	. . . . . . . . . 10 |- ( y = x -> ( g " ( w \ ( f " y ) ) ) = ( g " ( w \ ( f " x ) ) ) )
25		difeq2 3316	. . . . . . . . . 10 |- ( y = x -> ( z \ y ) = ( z \ x ) )
26	24, 25	sseq12d 3255	. . . . . . . . 9 |- ( y = x -> ( ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) <-> ( g " ( w \ ( f " x ) ) ) C_ ( z \ x ) ) )
27	21, 26	anbi12d 473	. . . . . . . 8 |- ( y = x -> ( ( y C_ z /\ ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) ) <-> ( x C_ z /\ ( g " ( w \ ( f " x ) ) ) C_ ( z \ x ) ) ) )
28	27	cbvabv 2354	. . . . . . 7 |- { y | ( y C_ z /\ ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) ) } = { x | ( x C_ z /\ ( g " ( w \ ( f " x ) ) ) C_ ( z \ x ) ) }
29		eqid 2229	. . . . . . 7 |- ( ( f |` U. { y | ( y C_ z /\ ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) ) } ) u. ( `' g |` ( z \ U. { y | ( y C_ z /\ ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) ) } ) ) ) = ( ( f |` U. { y | ( y C_ z /\ ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) ) } ) u. ( `' g |` ( z \ U. { y | ( y C_ z /\ ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) ) } ) ) )
30		vex 2802	. . . . . . 7 |- w e. _V
31	20, 28, 29, 30	sbthlemi10 7133	. . . . . 6 |- ( (EXMID /\ ( z ~<_ w /\ w ~<_ z ) ) -> z ~~ w )
32	31	ex 115	. . . . 5 |- (EXMID -> ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) )
33	32	adantr 276	. . . 4 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) )
34	13, 19, 33	vtocld 2853	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) )
35	5, 11, 34	vtocld 2853	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) )
36	1, 2, 35	mp2and 433	1 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   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   ~~ 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:  exmidsbth  16392