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Mirrors > Home > ILE Home > Th. List > exmidomniim | Unicode version |
Description: Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7097. (Contributed by Jim Kingdon, 29-Jun-2022.) |
Ref | Expression |
---|---|
exmidomniim | EXMID Omni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidexmid 4169 | . . . . . . . . 9 EXMID DECID | |
2 | exmiddc 826 | . . . . . . . . 9 DECID | |
3 | 1, 2 | syl 14 | . . . . . . . 8 EXMID |
4 | 3 | orcomd 719 | . . . . . . 7 EXMID |
5 | 4 | adantr 274 | . . . . . 6 EXMID |
6 | ffvelrn 5612 | . . . . . . . . . . . . . 14 | |
7 | df2o3 6389 | . . . . . . . . . . . . . 14 | |
8 | 6, 7 | eleqtrdi 2257 | . . . . . . . . . . . . 13 |
9 | elpri 3593 | . . . . . . . . . . . . 13 | |
10 | 8, 9 | syl 14 | . . . . . . . . . . . 12 |
11 | 10 | ord 714 | . . . . . . . . . . 11 |
12 | 11 | ralimdva 2531 | . . . . . . . . . 10 |
13 | 12 | con3d 621 | . . . . . . . . 9 |
14 | 13 | adantl 275 | . . . . . . . 8 EXMID |
15 | exmidexmid 4169 | . . . . . . . . . 10 EXMID DECID | |
16 | dfrex2dc 2455 | . . . . . . . . . 10 DECID | |
17 | 15, 16 | syl 14 | . . . . . . . . 9 EXMID |
18 | 17 | adantr 274 | . . . . . . . 8 EXMID |
19 | 14, 18 | sylibrd 168 | . . . . . . 7 EXMID |
20 | 19 | orim1d 777 | . . . . . 6 EXMID |
21 | 5, 20 | mpd 13 | . . . . 5 EXMID |
22 | 21 | ex 114 | . . . 4 EXMID |
23 | 22 | alrimiv 1861 | . . 3 EXMID |
24 | isomni 7091 | . . . 4 Omni | |
25 | 24 | elv 2725 | . . 3 Omni |
26 | 23, 25 | sylibr 133 | . 2 EXMID Omni |
27 | 26 | alrimiv 1861 | 1 EXMID Omni |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 824 wal 1340 wceq 1342 wcel 2135 wral 2442 wrex 2443 cvv 2721 c0 3404 cpr 3571 EXMIDwem 4167 wf 5178 cfv 5182 c1o 6368 c2o 6369 Omnicomni 7089 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-exmid 4168 df-id 4265 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-1o 6375 df-2o 6376 df-omni 7090 |
This theorem is referenced by: exmidomni 7097 |
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