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Mirrors > Home > ILE Home > Th. List > exmid1dc | Unicode version |
Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4166 or ordtriexmid 4492. In this context can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.) |
Ref | Expression |
---|---|
exmid1dc.x | DECID |
Ref | Expression |
---|---|
exmid1dc | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid1dc.x | . . . . . . 7 DECID | |
2 | exmiddc 826 | . . . . . . 7 DECID | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | df-ne 2335 | . . . . . . . . 9 | |
5 | pwntru 4172 | . . . . . . . . . 10 | |
6 | 5 | ex 114 | . . . . . . . . 9 |
7 | 4, 6 | syl5bir 152 | . . . . . . . 8 |
8 | 7 | orim2d 778 | . . . . . . 7 |
9 | 8 | adantl 275 | . . . . . 6 |
10 | 3, 9 | mpd 13 | . . . . 5 |
11 | 10 | orcomd 719 | . . . 4 |
12 | 11 | ex 114 | . . 3 |
13 | 12 | alrimiv 1861 | . 2 |
14 | exmid01 4171 | . 2 EXMID | |
15 | 13, 14 | sylibr 133 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 wal 1340 wceq 1342 wne 2334 wss 3111 c0 3404 csn 3570 EXMIDwem 4167 |
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-ext 2146 ax-nul 4102 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-exmid 4168 |
This theorem is referenced by: pw1fin 6867 exmidonfin 7141 exmidaclem 7155 exmidontri 7186 exmidontri2or 7190 |
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