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Description: Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set is smaller (has lower cardinality) than and vice-versa, then and 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 6853 through sbthlemi10 6862; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6862. 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 13393. (Contributed by NM, 8-Jun-1998.) |
Ref | Expression |
---|---|
isbth | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 521 | . 2 EXMID | |
2 | simprr 522 | . 2 EXMID | |
3 | reldom 6647 | . . . . 5 | |
4 | 3 | brrelex1i 4590 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 EXMID |
6 | breq2 3941 | . . . . . 6 | |
7 | breq1 3940 | . . . . . 6 | |
8 | 6, 7 | anbi12d 465 | . . . . 5 |
9 | breq2 3941 | . . . . 5 | |
10 | 8, 9 | imbi12d 233 | . . . 4 |
11 | 10 | adantl 275 | . . 3 EXMID |
12 | 3 | brrelex1i 4590 | . . . . 5 |
13 | 1, 12 | syl 14 | . . . 4 EXMID |
14 | breq1 3940 | . . . . . . 7 | |
15 | breq2 3941 | . . . . . . 7 | |
16 | 14, 15 | anbi12d 465 | . . . . . 6 |
17 | breq1 3940 | . . . . . 6 | |
18 | 16, 17 | imbi12d 233 | . . . . 5 |
19 | 18 | adantl 275 | . . . 4 EXMID |
20 | vex 2692 | . . . . . . 7 | |
21 | sseq1 3125 | . . . . . . . . 9 | |
22 | imaeq2 4885 | . . . . . . . . . . . 12 | |
23 | 22 | difeq2d 3199 | . . . . . . . . . . 11 |
24 | 23 | imaeq2d 4889 | . . . . . . . . . 10 |
25 | difeq2 3193 | . . . . . . . . . 10 | |
26 | 24, 25 | sseq12d 3133 | . . . . . . . . 9 |
27 | 21, 26 | anbi12d 465 | . . . . . . . 8 |
28 | 27 | cbvabv 2265 | . . . . . . 7 |
29 | eqid 2140 | . . . . . . 7 | |
30 | vex 2692 | . . . . . . 7 | |
31 | 20, 28, 29, 30 | sbthlemi10 6862 | . . . . . 6 EXMID |
32 | 31 | ex 114 | . . . . 5 EXMID |
33 | 32 | adantr 274 | . . . 4 EXMID |
34 | 13, 19, 33 | vtocld 2741 | . . 3 EXMID |
35 | 5, 11, 34 | vtocld 2741 | . 2 EXMID |
36 | 1, 2, 35 | mp2and 430 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1332 wcel 1481 cab 2126 cvv 2689 cdif 3073 cun 3074 wss 3076 cuni 3744 class class class wbr 3937 EXMIDwem 4126 ccnv 4546 cres 4549 cima 4550 cen 6640 cdom 6641 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-exmid 4127 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-en 6643 df-dom 6644 |
This theorem is referenced by: exmidsbth 13394 |
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