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Mirrors > Home > ILE Home > Th. List > fisbth | Unicode version |
Description: Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.) |
Ref | Expression |
---|---|
fisbth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6241 | . . . 4 | |
2 | 1 | biimpi 113 | . . 3 |
3 | 2 | ad2antrr 457 | . 2 |
4 | isfi 6241 | . . . . 5 | |
5 | 4 | biimpi 113 | . . . 4 |
6 | 5 | ad3antlr 462 | . . 3 |
7 | simplrr 488 | . . . . 5 | |
8 | 7 | ensymd 6263 | . . . . . . . . 9 |
9 | simprl 483 | . . . . . . . . . 10 | |
10 | 9 | ad2antrr 457 | . . . . . . . . 9 |
11 | endomtr 6270 | . . . . . . . . 9 | |
12 | 8, 10, 11 | syl2anc 391 | . . . . . . . 8 |
13 | simprr 484 | . . . . . . . 8 | |
14 | domentr 6271 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 391 | . . . . . . 7 |
16 | simplrl 487 | . . . . . . . 8 | |
17 | simprl 483 | . . . . . . . 8 | |
18 | nndomo 6326 | . . . . . . . 8 | |
19 | 16, 17, 18 | syl2anc 391 | . . . . . . 7 |
20 | 15, 19 | mpbid 135 | . . . . . 6 |
21 | 13 | ensymd 6263 | . . . . . . . . 9 |
22 | simprr 484 | . . . . . . . . . 10 | |
23 | 22 | ad2antrr 457 | . . . . . . . . 9 |
24 | endomtr 6270 | . . . . . . . . 9 | |
25 | 21, 23, 24 | syl2anc 391 | . . . . . . . 8 |
26 | domentr 6271 | . . . . . . . 8 | |
27 | 25, 7, 26 | syl2anc 391 | . . . . . . 7 |
28 | nndomo 6326 | . . . . . . . 8 | |
29 | 17, 16, 28 | syl2anc 391 | . . . . . . 7 |
30 | 27, 29 | mpbid 135 | . . . . . 6 |
31 | 20, 30 | eqssd 2962 | . . . . 5 |
32 | 7, 31 | breqtrd 3788 | . . . 4 |
33 | entr 6264 | . . . 4 | |
34 | 32, 21, 33 | syl2anc 391 | . . 3 |
35 | 6, 34 | rexlimddv 2437 | . 2 |
36 | 3, 35 | rexlimddv 2437 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wcel 1393 wrex 2307 wss 2917 class class class wbr 3764 com 4313 cen 6219 cdom 6220 cfn 6221 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-er 6106 df-en 6222 df-dom 6223 df-fin 6224 |
This theorem is referenced by: (None) |
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