Theorem isbth 7095

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 7085 through sbthlemi10 7094; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7094. 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 16164. (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 6855	. . . . 5 |- Rel ~<_
4	3	brrelex1i 4736	. . . 4 |- ( B ~<_ A -> B e. _V )
5	2, 4	syl 14	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> B e. _V )
6		breq2 4063	. . . . . 6 |- ( w = B -> ( A ~<_ w <-> A ~<_ B ) )
7		breq1 4062	. . . . . 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 4063	. . . . 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 4736	. . . . 5 |- ( A ~<_ B -> A e. _V )
13	1, 12	syl 14	. . . 4 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A e. _V )
14		breq1 4062	. . . . . . 7 |- ( z = A -> ( z ~<_ w <-> A ~<_ w ) )
15		breq2 4063	. . . . . . 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 4062	. . . . . 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 2779	. . . . . . 7 |- z e. _V
21		sseq1 3224	. . . . . . . . 9 |- ( y = x -> ( y C_ z <-> x C_ z ) )
22		imaeq2 5037	. . . . . . . . . . . 12 |- ( y = x -> ( f " y ) = ( f " x ) )
23	22	difeq2d 3299	. . . . . . . . . . 11 |- ( y = x -> ( w \ ( f " y ) ) = ( w \ ( f " x ) ) )
24	23	imaeq2d 5041	. . . . . . . . . 10 |- ( y = x -> ( g " ( w \ ( f " y ) ) ) = ( g " ( w \ ( f " x ) ) ) )
25		difeq2 3293	. . . . . . . . . 10 |- ( y = x -> ( z \ y ) = ( z \ x ) )
26	24, 25	sseq12d 3232	. . . . . . . . 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 2332	. . . . . . 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 2207	. . . . . . 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 2779	. . . . . . 7 |- w e. _V
31	20, 28, 29, 30	sbthlemi10 7094	. . . . . 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 2830	. . 3 |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> ( ( A ~<_ w /\ w ~<_ A ) -> A ~~ w ) )
35	5, 11, 34	vtocld 2830	. 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 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   \ cdif 3171   u. cun 3172   C_ wss 3174   U.cuni 3864   class class class wbr 4059  EXMIDwem 4254   `'ccnv 4692   |` cres 4695   "cima 4696   ~~ cen 6848   ~<_ cdom 6849

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-exmid 4255  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-en 6851  df-dom 6852

This theorem is referenced by:  exmidsbth  16165