Theorem isbth 7289

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 7279 through sbthlemi10 7288; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7288. 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 17274. (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 535	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~<_ B )
2		simprr 537	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> B ~<_ A )
3		reldom 7031	. . . . 5 |- Rel ~<_
4	3	brrelex1i 4820	. . . 4 |- ( B ~<_ A -> B e. _V )
5	2, 4	syl 14	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> B e. _V )
6		breq2 4135	. . . . . 6 |- ( w = B -> ( A ~<_ w <-> A ~<_ B ) )
7		breq1 4134	. . . . . 6 |- ( w = B -> ( w ~<_ A <-> B ~<_ A ) )
8	6, 7	anbi12d 477	. . . . 5 |- ( w = B -> ( ( A ~<_ w /\ w ~<_ A ) <-> ( A ~<_ B /\ B ~<_ A ) ) )
9		breq2 4135	. . . . 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 4820	. . . . 5 |- ( A ~<_ B -> A e. _V )
13	1, 12	syl 14	. . . 4 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A e. _V )
14		breq1 4134	. . . . . . 7 |- ( z = A -> ( z ~<_ w <-> A ~<_ w ) )
15		breq2 4135	. . . . . . 7 |- ( z = A -> ( w ~<_ z <-> w ~<_ A ) )
16	14, 15	anbi12d 477	. . . . . 6 |- ( z = A -> ( ( z ~<_ w /\ w ~<_ z ) <-> ( A ~<_ w /\ w ~<_ A ) ) )
17		breq1 4134	. . . . . 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 2824	. . . . . . 7 |- z e. _V
21		sseq1 3271	. . . . . . . . 9 |- ( y = x -> ( y C_ z <-> x C_ z ) )
22		imaeq2 5125	. . . . . . . . . . . 12 |- ( y = x -> ( f " y ) = ( f " x ) )
23	22	difeq2d 3347	. . . . . . . . . . 11 |- ( y = x -> ( w \ ( f " y ) ) = ( w \ ( f " x ) ) )
24	23	imaeq2d 5129	. . . . . . . . . 10 |- ( y = x -> ( g " ( w \ ( f " y ) ) ) = ( g " ( w \ ( f " x ) ) ) )
25		difeq2 3341	. . . . . . . . . 10 |- ( y = x -> ( z \ y ) = ( z \ x ) )
26	24, 25	sseq12d 3279	. . . . . . . . 9 |- ( y = x -> ( ( g " ( w \ ( f " y ) ) ) C_ ( z \ y ) <-> ( g " ( w \ ( f " x ) ) ) C_ ( z \ x ) ) )
27	21, 26	anbi12d 477	. . . . . . . 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 2365	. . . . . . 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 2238	. . . . . . 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 2824	. . . . . . 7 |- w e. _V
31	20, 28, 29, 30	sbthlemi10 7288	. . . . . 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 2875	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) )
35	5, 11, 34	vtocld 2875	. 2 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) )
36	1, 2, 35	mp2and 437	1 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   \ cdif 3217   u. cun 3218   C_ wss 3220   U.cuni 3935   class class class wbr 4131  EXMIDwem 4333   `'ccnv 4775   |` cres 4778   "cima 4779   ~~ cen 7024   ~<_ cdom 7025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-exmid 4334  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-en 7027  df-dom 7028

This theorem is used by:  exmidsbth  17275