Mathbox for Jim Kingdon |
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Description: Schroeder-Bernstein is not possible even for . We know by exmidsbth 13219 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.) |
Ref | Expression |
---|---|
sbthom | Omni EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p0ex 4112 | . . . . . . . . . . 11 | |
2 | 1 | ssex 4065 | . . . . . . . . . 10 |
3 | 2 | adantl 275 | . . . . . . . . 9 Omni |
4 | omex 4507 | . . . . . . . . 9 | |
5 | djuex 6928 | . . . . . . . . 9 ⊔ | |
6 | 3, 4, 5 | sylancl 409 | . . . . . . . 8 Omni ⊔ |
7 | simpll 518 | . . . . . . . 8 Omni | |
8 | ssdomg 6672 | . . . . . . . . . . . 12 | |
9 | 1, 8 | ax-mp 5 | . . . . . . . . . . 11 |
10 | domrefg 6661 | . . . . . . . . . . . . . 14 | |
11 | 4, 10 | ax-mp 5 | . . . . . . . . . . . . 13 |
12 | djudom 6978 | . . . . . . . . . . . . 13 ⊔ ⊔ | |
13 | 11, 12 | mpan2 421 | . . . . . . . . . . . 12 ⊔ ⊔ |
14 | df1o2 6326 | . . . . . . . . . . . . 13 | |
15 | djueq1 6925 | . . . . . . . . . . . . 13 ⊔ ⊔ | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . . . . 12 ⊔ ⊔ |
17 | 13, 16 | breqtrrdi 3970 | . . . . . . . . . . 11 ⊔ ⊔ |
18 | 1onn 6416 | . . . . . . . . . . . . . 14 | |
19 | endjusym 6981 | . . . . . . . . . . . . . 14 ⊔ ⊔ | |
20 | 4, 18, 19 | mp2an 422 | . . . . . . . . . . . . 13 ⊔ ⊔ |
21 | omp1eom 6980 | . . . . . . . . . . . . 13 ⊔ | |
22 | 20, 21 | entr3i 6682 | . . . . . . . . . . . 12 ⊔ |
23 | domentr 6685 | . . . . . . . . . . . 12 ⊔ ⊔ ⊔ ⊔ | |
24 | 22, 23 | mpan2 421 | . . . . . . . . . . 11 ⊔ ⊔ ⊔ |
25 | 9, 17, 24 | 3syl 17 | . . . . . . . . . 10 ⊔ |
26 | 25 | adantl 275 | . . . . . . . . 9 Omni ⊔ |
27 | djudomr 7076 | . . . . . . . . . 10 ⊔ | |
28 | 3, 4, 27 | sylancl 409 | . . . . . . . . 9 Omni ⊔ |
29 | 26, 28 | jca 304 | . . . . . . . 8 Omni ⊔ ⊔ |
30 | breq1 3932 | . . . . . . . . . . 11 ⊔ ⊔ | |
31 | breq2 3933 | . . . . . . . . . . 11 ⊔ ⊔ | |
32 | 30, 31 | anbi12d 464 | . . . . . . . . . 10 ⊔ ⊔ ⊔ |
33 | breq1 3932 | . . . . . . . . . 10 ⊔ ⊔ | |
34 | 32, 33 | imbi12d 233 | . . . . . . . . 9 ⊔ ⊔ ⊔ ⊔ |
35 | 34 | spcgv 2773 | . . . . . . . 8 ⊔ ⊔ ⊔ ⊔ |
36 | 6, 7, 29, 35 | syl3c 63 | . . . . . . 7 Omni ⊔ |
37 | 36 | ensymd 6677 | . . . . . 6 Omni ⊔ |
38 | bren 6641 | . . . . . 6 ⊔ ⊔ | |
39 | 37, 38 | sylib 121 | . . . . 5 Omni ⊔ |
40 | simpllr 523 | . . . . . 6 Omni ⊔ Omni | |
41 | simplr 519 | . . . . . 6 Omni ⊔ | |
42 | simpr 109 | . . . . . 6 Omni ⊔ ⊔ | |
43 | 40, 41, 42 | sbthomlem 13220 | . . . . 5 Omni ⊔ |
44 | 39, 43 | exlimddv 1870 | . . . 4 Omni |
45 | 44 | ex 114 | . . 3 Omni |
46 | 45 | alrimiv 1846 | . 2 Omni |
47 | exmid01 4121 | . 2 EXMID | |
48 | 46, 47 | sylibr 133 | 1 Omni EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wal 1329 wceq 1331 wex 1468 wcel 1480 cvv 2686 wss 3071 c0 3363 csn 3527 class class class wbr 3929 EXMIDwem 4118 com 4504 wf1o 5122 c1o 6306 cen 6632 cdom 6633 ⊔ cdju 6922 Omnicomni 7004 |
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-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-exmid 4119 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 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-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-1o 6313 df-2o 6314 df-er 6429 df-map 6544 df-en 6635 df-dom 6636 df-dju 6923 df-inl 6932 df-inr 6933 df-case 6969 df-omni 7006 |
This theorem is referenced by: (None) |
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