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Mirrors > Home > ILE Home > Th. List > exmid01 | Unicode version |
Description: Excluded middle is equivalent to saying any subset of is either or . (Contributed by BJ and Jim Kingdon, 18-Jun-2022.) |
Ref | Expression |
---|---|
exmid01 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-exmid 4114 | . 2 EXMID DECID | |
2 | df-dc 820 | . . . . 5 DECID | |
3 | orcom 717 | . . . . . 6 | |
4 | simpll 518 | . . . . . . . . . . . . . 14 | |
5 | simpr 109 | . . . . . . . . . . . . . 14 | |
6 | 4, 5 | sseldd 3093 | . . . . . . . . . . . . 13 |
7 | velsn 3539 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | sylib 121 | . . . . . . . . . . . 12 |
9 | 8, 5 | eqeltrrd 2215 | . . . . . . . . . . 11 |
10 | simplr 519 | . . . . . . . . . . 11 | |
11 | 9, 10 | pm2.65da 650 | . . . . . . . . . 10 |
12 | 11 | eq0rdv 3402 | . . . . . . . . 9 |
13 | 12 | ex 114 | . . . . . . . 8 |
14 | noel 3362 | . . . . . . . . 9 | |
15 | eleq2 2201 | . . . . . . . . 9 | |
16 | 14, 15 | mtbiri 664 | . . . . . . . 8 |
17 | 13, 16 | impbid1 141 | . . . . . . 7 |
18 | elex2 2697 | . . . . . . . . . 10 | |
19 | sssnm 3676 | . . . . . . . . . 10 | |
20 | 18, 19 | syl 14 | . . . . . . . . 9 |
21 | 20 | biimpcd 158 | . . . . . . . 8 |
22 | 0ex 4050 | . . . . . . . . . 10 | |
23 | 22 | snid 3551 | . . . . . . . . 9 |
24 | eleq2 2201 | . . . . . . . . 9 | |
25 | 23, 24 | mpbiri 167 | . . . . . . . 8 |
26 | 21, 25 | impbid1 141 | . . . . . . 7 |
27 | 17, 26 | orbi12d 782 | . . . . . 6 |
28 | 3, 27 | syl5bb 191 | . . . . 5 |
29 | 2, 28 | syl5bb 191 | . . . 4 DECID |
30 | 29 | pm5.74i 179 | . . 3 DECID |
31 | 30 | albii 1446 | . 2 DECID |
32 | 1, 31 | bitri 183 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 wal 1329 wceq 1331 wex 1468 wcel 1480 wss 3066 c0 3358 csn 3522 EXMIDwem 4113 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-nul 4049 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-exmid 4114 |
This theorem is referenced by: exmid1dc 4118 exmidn0m 4119 exmidsssn 4120 exmidpw 6795 exmidomni 7007 ss1oel2o 13178 exmidsbthrlem 13206 sbthom 13210 |
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