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Mirrors > Home > ILE Home > Th. List > ctssexmid | Unicode version |
Description: The decidability condition in ctssdc 6998 is needed. More specifically, ctssdc 6998 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.) |
Ref | Expression |
---|---|
ctssexmid.1 | ⊔ |
ctssexmid.lpo | Omni |
Ref | Expression |
---|---|
ctssexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3182 | . . 3 | |
2 | f1oi 5405 | . . . 4 | |
3 | f1ofo 5374 | . . . 4 | |
4 | ctssexmid.lpo | . . . . . . . 8 Omni | |
5 | 4 | elexi 2698 | . . . . . . 7 |
6 | 5 | rabex 4072 | . . . . . 6 |
7 | resiexg 4864 | . . . . . 6 | |
8 | 6, 7 | ax-mp 5 | . . . . 5 |
9 | foeq1 5341 | . . . . 5 | |
10 | 8, 9 | spcev 2780 | . . . 4 |
11 | 2, 3, 10 | mp2b 8 | . . 3 |
12 | simpr 109 | . . . . . . 7 | |
13 | 12 | sseq1d 3126 | . . . . . 6 |
14 | eqidd 2140 | . . . . . . . 8 | |
15 | simpl 108 | . . . . . . . 8 | |
16 | 14, 12, 15 | foeq123d 5361 | . . . . . . 7 |
17 | 16 | exbidv 1797 | . . . . . 6 |
18 | 13, 17 | anbi12d 464 | . . . . 5 |
19 | djueq1 6925 | . . . . . . 7 ⊔ ⊔ | |
20 | foeq3 5343 | . . . . . . 7 ⊔ ⊔ ⊔ ⊔ | |
21 | 15, 19, 20 | 3syl 17 | . . . . . 6 ⊔ ⊔ |
22 | 21 | exbidv 1797 | . . . . 5 ⊔ ⊔ |
23 | 18, 22 | imbi12d 233 | . . . 4 ⊔ ⊔ |
24 | ctssexmid.1 | . . . 4 ⊔ | |
25 | 6, 6, 23, 24 | vtocl2 2741 | . . 3 ⊔ |
26 | 1, 11, 25 | mp2an 422 | . 2 ⊔ |
27 | 4 | a1i 9 | . . . 4 ⊔ Omni |
28 | id 19 | . . . 4 ⊔ ⊔ | |
29 | 27, 28 | fodjuomni 7021 | . . 3 ⊔ |
30 | 29 | exlimiv 1577 | . 2 ⊔ |
31 | biidd 171 | . . . . . 6 | |
32 | 31 | elrab 2840 | . . . . 5 |
33 | 32 | simprbi 273 | . . . 4 |
34 | 33 | exlimiv 1577 | . . 3 |
35 | rabeq0 3392 | . . . 4 | |
36 | peano1 4508 | . . . . 5 | |
37 | elex2 2702 | . . . . 5 | |
38 | r19.3rmv 3453 | . . . . 5 | |
39 | 36, 37, 38 | mp2b 8 | . . . 4 |
40 | 35, 39 | sylbb2 137 | . . 3 |
41 | 34, 40 | orim12i 748 | . 2 |
42 | 26, 30, 41 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 wral 2416 crab 2420 cvv 2686 wss 3071 c0 3363 cid 4210 com 4504 cres 4541 wfo 5121 wf1o 5122 c1o 6306 ⊔ 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-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
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-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-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 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-map 6544 df-dju 6923 df-inl 6932 df-inr 6933 df-omni 7006 |
This theorem is referenced by: (None) |
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