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Mirrors > Home > MPE Home > Th. List > sbth | Structured version Visualization version Unicode version |
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 8070 through sbthlem10 8079; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 8079. 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. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.) |
Ref | Expression |
---|---|
sbth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 7961 | . . . 4 | |
2 | 1 | brrelexi 5158 | . . 3 |
3 | 1 | brrelexi 5158 | . . 3 |
4 | breq1 4656 | . . . . . 6 | |
5 | breq2 4657 | . . . . . 6 | |
6 | 4, 5 | anbi12d 747 | . . . . 5 |
7 | breq1 4656 | . . . . 5 | |
8 | 6, 7 | imbi12d 334 | . . . 4 |
9 | breq2 4657 | . . . . . 6 | |
10 | breq1 4656 | . . . . . 6 | |
11 | 9, 10 | anbi12d 747 | . . . . 5 |
12 | breq2 4657 | . . . . 5 | |
13 | 11, 12 | imbi12d 334 | . . . 4 |
14 | vex 3203 | . . . . 5 | |
15 | sseq1 3626 | . . . . . . 7 | |
16 | imaeq2 5462 | . . . . . . . . . 10 | |
17 | 16 | difeq2d 3728 | . . . . . . . . 9 |
18 | 17 | imaeq2d 5466 | . . . . . . . 8 |
19 | difeq2 3722 | . . . . . . . 8 | |
20 | 18, 19 | sseq12d 3634 | . . . . . . 7 |
21 | 15, 20 | anbi12d 747 | . . . . . 6 |
22 | 21 | cbvabv 2747 | . . . . 5 |
23 | eqid 2622 | . . . . 5 | |
24 | vex 3203 | . . . . 5 | |
25 | 14, 22, 23, 24 | sbthlem10 8079 | . . . 4 |
26 | 8, 13, 25 | vtocl2g 3270 | . . 3 |
27 | 2, 3, 26 | syl2an 494 | . 2 |
28 | 27 | pm2.43i 52 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cab 2608 cvv 3200 cdif 3571 cun 3572 wss 3574 cuni 4436 class class class wbr 4653 ccnv 5113 cres 5116 cima 5117 cen 7952 cdom 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-en 7956 df-dom 7957 |
This theorem is referenced by: sbthb 8081 sdomnsym 8085 domtriord 8106 xpen 8123 limenpsi 8135 php 8144 onomeneq 8150 unbnn 8216 infxpenlem 8836 fseqen 8850 infpwfien 8885 inffien 8886 alephdom 8904 mappwen 8935 infcdaabs 9028 infunabs 9029 infcda 9030 infdif 9031 infxpabs 9034 infmap2 9040 gchaleph 9493 gchhar 9501 inttsk 9596 inar1 9597 znnen 14941 qnnen 14942 rpnnen 14956 rexpen 14957 mreexfidimd 16311 acsinfdimd 17182 fislw 18040 opnreen 22634 ovolctb2 23260 vitali 23382 aannenlem3 24085 basellem4 24810 lgsqrlem4 25074 upgrex 25987 phpreu 33393 poimirlem26 33435 pellexlem4 37396 pellexlem5 37397 idomsubgmo 37776 |
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