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Mirrors > Home > ILE Home > Th. List > ctssexmid | Unicode version |
Description: The decidability condition in ctssdc 7006 is needed. More specifically, ctssdc 7006 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 3187 | . . 3 | |
2 | f1oi 5413 | . . . 4 | |
3 | f1ofo 5382 | . . . 4 | |
4 | ctssexmid.lpo | . . . . . . . 8 Omni | |
5 | 4 | elexi 2701 | . . . . . . 7 |
6 | 5 | rabex 4080 | . . . . . 6 |
7 | resiexg 4872 | . . . . . 6 | |
8 | 6, 7 | ax-mp 5 | . . . . 5 |
9 | foeq1 5349 | . . . . 5 | |
10 | 8, 9 | spcev 2784 | . . . 4 |
11 | 2, 3, 10 | mp2b 8 | . . 3 |
12 | simpr 109 | . . . . . . 7 | |
13 | 12 | sseq1d 3131 | . . . . . 6 |
14 | eqidd 2141 | . . . . . . . 8 | |
15 | simpl 108 | . . . . . . . 8 | |
16 | 14, 12, 15 | foeq123d 5369 | . . . . . . 7 |
17 | 16 | exbidv 1798 | . . . . . 6 |
18 | 13, 17 | anbi12d 465 | . . . . 5 |
19 | djueq1 6933 | . . . . . . 7 ⊔ ⊔ | |
20 | foeq3 5351 | . . . . . . 7 ⊔ ⊔ ⊔ ⊔ | |
21 | 15, 19, 20 | 3syl 17 | . . . . . 6 ⊔ ⊔ |
22 | 21 | exbidv 1798 | . . . . 5 ⊔ ⊔ |
23 | 18, 22 | imbi12d 233 | . . . 4 ⊔ ⊔ |
24 | ctssexmid.1 | . . . 4 ⊔ | |
25 | 6, 6, 23, 24 | vtocl2 2744 | . . 3 ⊔ |
26 | 1, 11, 25 | mp2an 423 | . 2 ⊔ |
27 | 4 | a1i 9 | . . . 4 ⊔ Omni |
28 | id 19 | . . . 4 ⊔ ⊔ | |
29 | 27, 28 | fodjuomni 7029 | . . 3 ⊔ |
30 | 29 | exlimiv 1578 | . 2 ⊔ |
31 | biidd 171 | . . . . . 6 | |
32 | 31 | elrab 2844 | . . . . 5 |
33 | 32 | simprbi 273 | . . . 4 |
34 | 33 | exlimiv 1578 | . . 3 |
35 | rabeq0 3397 | . . . 4 | |
36 | peano1 4516 | . . . . 5 | |
37 | elex2 2705 | . . . . 5 | |
38 | r19.3rmv 3458 | . . . . 5 | |
39 | 36, 37, 38 | mp2b 8 | . . . 4 |
40 | 35, 39 | sylbb2 137 | . . 3 |
41 | 34, 40 | orim12i 749 | . 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 698 wceq 1332 wex 1469 wcel 1481 wral 2417 crab 2421 cvv 2689 wss 3076 c0 3368 cid 4218 com 4512 cres 4549 wfo 5129 wf1o 5130 c1o 6314 ⊔ cdju 6930 Omnicomni 7012 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-1o 6321 df-2o 6322 df-map 6552 df-dju 6931 df-inl 6940 df-inr 6941 df-omni 7014 |
This theorem is referenced by: (None) |
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