Theorem isbth 6908

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 6898 through sbthlemi10 6907; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6907. 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 13565. (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 521	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~<_ B )
2		simprr 522	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> B ~<_ A )
3		reldom 6687	. . . . 5 |- Rel ~<_
4	3	brrelex1i 4628	. . . 4 |- ( B ~<_ A -> B e. _V )
5	2, 4	syl 14	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> B e. _V )
6		breq2 3969	. . . . . 6 |- ( w = B -> ( A ~<_ w <-> A ~<_ B ) )
7		breq1 3968	. . . . . 6 |- ( w = B -> ( w ~<_ A <-> B ~<_ A ) )
8	6, 7	anbi12d 465	. . . . 5 |- ( w = B -> ( ( A ~<_ w /\ w ~<_ A ) <-> ( A ~<_ B /\ B ~<_ A ) ) )
9		breq2 3969	. . . . 5 |- ( w = B -> ( A ~~ w <-> A ~~ B ) )
10	8, 9	imbi12d 233	. . . 4 |- ( w = B -> ( ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) <-> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) ) )
11	10	adantl 275	. . 3 |- ( ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) /\ w = B ) -> ( ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) <-> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) ) )
12	3	brrelex1i 4628	. . . . 5 |- ( A ~<_ B -> A e. _V )
13	1, 12	syl 14	. . . 4 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A e. _V )
14		breq1 3968	. . . . . . 7 |- ( z = A -> ( z ~<_ w <-> A ~<_ w ) )
15		breq2 3969	. . . . . . 7 |- ( z = A -> ( w ~<_ z <-> w ~<_ A ) )
16	14, 15	anbi12d 465	. . . . . 6 |- ( z = A -> ( ( z ~<_ w /\ w ~<_ z ) <-> ( A ~<_ w /\ w ~<_ A ) ) )
17		breq1 3968	. . . . . 6 |- ( z = A -> ( z ~~ w <-> A ~~ w ) )
18	16, 17	imbi12d 233	. . . . 5 |- ( z = A -> ( ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) <-> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) ) )
19	18	adantl 275	. . . 4 |- ( ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) /\ z = A ) -> ( ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) <-> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) ) )
20		vex 2715	. . . . . . 7 |- z e. _V
21		sseq1 3151	. . . . . . . . 9 |- ( y = x -> ( y C_ z <-> x C_ z ) )
22		imaeq2 4923	. . . . . . . . . . . 12 |- ( y = x -> ( f " y ) = ( f " x ) )
23	22	difeq2d 3225	. . . . . . . . . . 11 |- ( y = x -> ( w \ ( f " y ) ) = ( w \ ( f " x ) ) )
24	23	imaeq2d 4927	. . . . . . . . . 10 |- ( y = x -> ( g " ( w \ ( f " y ) ) ) = ( g " ( w \ ( f " x ) ) ) )
25		difeq2 3219	. . . . . . . . . 10 |- ( y = x -> ( z \ y ) = ( z \ x ) )
26	24, 25	sseq12d 3159	. . . . . . . . 9 |- ( y = x -> ( ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) <-> ( g " ( w \ ( f " x ) ) ) C_ ( z \ x ) ) )
27	21, 26	anbi12d 465	. . . . . . . 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 2282	. . . . . . 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 2157	. . . . . . 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 2715	. . . . . . 7 |- w e. _V
31	20, 28, 29, 30	sbthlemi10 6907	. . . . . 6 |- ( (EXMID /\ ( z ~<_ w /\ w ~<_ z ) ) -> z ~~ w )
32	31	ex 114	. . . . 5 |- (EXMID -> ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) )
33	32	adantr 274	. . . 4 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( z ~<_ w /\ w ~<_ z ) -> z ~~ w ) )
34	13, 19, 33	vtocld 2764	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) )
35	5, 11, 34	vtocld 2764	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) )
36	1, 2, 35	mp2and 430	1 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   {cab 2143   _Vcvv 2712   \ cdif 3099   u. cun 3100   C_ wss 3102   U.cuni 3772   class class class wbr 3965  EXMIDwem 4155   `'ccnv 4584   |` cres 4587   "cima 4588   ~~ cen 6680   ~<_ cdom 6681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-exmid 4156  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-en 6683  df-dom 6684

This theorem is referenced by:  exmidsbth  13566