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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 6845 through sbthlemi10 6854; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6854. 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 13218. (Contributed by NM, 8-Jun-1998.) |
Ref | Expression |
---|---|
isbth | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 520 | . 2 EXMID | |
2 | simprr 521 | . 2 EXMID | |
3 | reldom 6639 | . . . . 5 | |
4 | 3 | brrelex1i 4582 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 EXMID |
6 | breq2 3933 | . . . . . 6 | |
7 | breq1 3932 | . . . . . 6 | |
8 | 6, 7 | anbi12d 464 | . . . . 5 |
9 | breq2 3933 | . . . . 5 | |
10 | 8, 9 | imbi12d 233 | . . . 4 |
11 | 10 | adantl 275 | . . 3 EXMID |
12 | 3 | brrelex1i 4582 | . . . . 5 |
13 | 1, 12 | syl 14 | . . . 4 EXMID |
14 | breq1 3932 | . . . . . . 7 | |
15 | breq2 3933 | . . . . . . 7 | |
16 | 14, 15 | anbi12d 464 | . . . . . 6 |
17 | breq1 3932 | . . . . . 6 | |
18 | 16, 17 | imbi12d 233 | . . . . 5 |
19 | 18 | adantl 275 | . . . 4 EXMID |
20 | vex 2689 | . . . . . . 7 | |
21 | sseq1 3120 | . . . . . . . . 9 | |
22 | imaeq2 4877 | . . . . . . . . . . . 12 | |
23 | 22 | difeq2d 3194 | . . . . . . . . . . 11 |
24 | 23 | imaeq2d 4881 | . . . . . . . . . 10 |
25 | difeq2 3188 | . . . . . . . . . 10 | |
26 | 24, 25 | sseq12d 3128 | . . . . . . . . 9 |
27 | 21, 26 | anbi12d 464 | . . . . . . . 8 |
28 | 27 | cbvabv 2264 | . . . . . . 7 |
29 | eqid 2139 | . . . . . . 7 | |
30 | vex 2689 | . . . . . . 7 | |
31 | 20, 28, 29, 30 | sbthlemi10 6854 | . . . . . 6 EXMID |
32 | 31 | ex 114 | . . . . 5 EXMID |
33 | 32 | adantr 274 | . . . 4 EXMID |
34 | 13, 19, 33 | vtocld 2738 | . . 3 EXMID |
35 | 5, 11, 34 | vtocld 2738 | . 2 EXMID |
36 | 1, 2, 35 | mp2and 429 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cab 2125 cvv 2686 cdif 3068 cun 3069 wss 3071 cuni 3736 class class class wbr 3929 EXMIDwem 4118 ccnv 4538 cres 4541 cima 4542 cen 6632 cdom 6633 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-exmid 4119 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-en 6635 df-dom 6636 |
This theorem is referenced by: exmidsbth 13219 |
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