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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 7118. (Contributed by Jim Kingdon, 29-Jun-2022.) |
Ref | Expression |
---|---|
exmidomniim | EXMID Omni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidexmid 4182 | . . . . . . . . 9 EXMID DECID | |
2 | exmiddc 831 | . . . . . . . . 9 DECID | |
3 | 1, 2 | syl 14 | . . . . . . . 8 EXMID |
4 | 3 | orcomd 724 | . . . . . . 7 EXMID |
5 | 4 | adantr 274 | . . . . . 6 EXMID |
6 | ffvelrn 5629 | . . . . . . . . . . . . . 14 | |
7 | df2o3 6409 | . . . . . . . . . . . . . 14 | |
8 | 6, 7 | eleqtrdi 2263 | . . . . . . . . . . . . 13 |
9 | elpri 3606 | . . . . . . . . . . . . 13 | |
10 | 8, 9 | syl 14 | . . . . . . . . . . . 12 |
11 | 10 | ord 719 | . . . . . . . . . . 11 |
12 | 11 | ralimdva 2537 | . . . . . . . . . 10 |
13 | 12 | con3d 626 | . . . . . . . . 9 |
14 | 13 | adantl 275 | . . . . . . . 8 EXMID |
15 | exmidexmid 4182 | . . . . . . . . . 10 EXMID DECID | |
16 | dfrex2dc 2461 | . . . . . . . . . 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 782 | . . . . . 6 EXMID |
21 | 5, 20 | mpd 13 | . . . . 5 EXMID |
22 | 21 | ex 114 | . . . 4 EXMID |
23 | 22 | alrimiv 1867 | . . 3 EXMID |
24 | isomni 7112 | . . . 4 Omni | |
25 | 24 | elv 2734 | . . 3 Omni |
26 | 23, 25 | sylibr 133 | . 2 EXMID Omni |
27 | 26 | alrimiv 1867 | 1 EXMID Omni |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 DECID wdc 829 wal 1346 wceq 1348 wcel 2141 wral 2448 wrex 2449 cvv 2730 c0 3414 cpr 3584 EXMIDwem 4180 wf 5194 cfv 5198 c1o 6388 c2o 6389 Omnicomni 7110 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-exmid 4181 df-id 4278 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-1o 6395 df-2o 6396 df-omni 7111 |
This theorem is referenced by: exmidomni 7118 |
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