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Mirrors > Home > ILE Home > Th. List > exmidundifim | Unicode version |
Description: Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4129 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.) |
Ref | Expression |
---|---|
exmidundifim | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undifss 3443 | . . . . . . 7 | |
2 | 1 | biimpi 119 | . . . . . 6 |
3 | 2 | adantl 275 | . . . . 5 EXMID |
4 | elun1 3243 | . . . . . . . . . 10 | |
5 | 4 | adantl 275 | . . . . . . . . 9 EXMID |
6 | simplr 519 | . . . . . . . . . . 11 EXMID | |
7 | simpr 109 | . . . . . . . . . . 11 EXMID | |
8 | 6, 7 | eldifd 3081 | . . . . . . . . . 10 EXMID |
9 | elun2 3244 | . . . . . . . . . 10 | |
10 | 8, 9 | syl 14 | . . . . . . . . 9 EXMID |
11 | exmidexmid 4120 | . . . . . . . . . . 11 EXMID DECID | |
12 | exmiddc 821 | . . . . . . . . . . 11 DECID | |
13 | 11, 12 | syl 14 | . . . . . . . . . 10 EXMID |
14 | 13 | adantr 274 | . . . . . . . . 9 EXMID |
15 | 5, 10, 14 | mpjaodan 787 | . . . . . . . 8 EXMID |
16 | 15 | ex 114 | . . . . . . 7 EXMID |
17 | 16 | ssrdv 3103 | . . . . . 6 EXMID |
18 | 17 | adantr 274 | . . . . 5 EXMID |
19 | 3, 18 | eqssd 3114 | . . . 4 EXMID |
20 | 19 | ex 114 | . . 3 EXMID |
21 | 20 | alrimivv 1847 | . 2 EXMID |
22 | vex 2689 | . . . . . 6 | |
23 | p0ex 4112 | . . . . . 6 | |
24 | sseq12 3122 | . . . . . . . 8 | |
25 | simpl 108 | . . . . . . . . . 10 | |
26 | simpr 109 | . . . . . . . . . . 11 | |
27 | 26, 25 | difeq12d 3195 | . . . . . . . . . 10 |
28 | 25, 27 | uneq12d 3231 | . . . . . . . . 9 |
29 | 28, 26 | eqeq12d 2154 | . . . . . . . 8 |
30 | 24, 29 | imbi12d 233 | . . . . . . 7 |
31 | 30 | spc2gv 2776 | . . . . . 6 |
32 | 22, 23, 31 | mp2an 422 | . . . . 5 |
33 | 0ex 4055 | . . . . . . . 8 | |
34 | 33 | snid 3556 | . . . . . . 7 |
35 | eleq2 2203 | . . . . . . 7 | |
36 | 34, 35 | mpbiri 167 | . . . . . 6 |
37 | eldifn 3199 | . . . . . . . 8 | |
38 | 37 | orim2i 750 | . . . . . . 7 |
39 | elun 3217 | . . . . . . 7 | |
40 | df-dc 820 | . . . . . . 7 DECID | |
41 | 38, 39, 40 | 3imtr4i 200 | . . . . . 6 DECID |
42 | 36, 41 | syl 14 | . . . . 5 DECID |
43 | 32, 42 | syl6 33 | . . . 4 DECID |
44 | 43 | alrimiv 1846 | . . 3 DECID |
45 | df-exmid 4119 | . . 3 EXMID DECID | |
46 | 44, 45 | sylibr 133 | . 2 EXMID |
47 | 21, 46 | impbii 125 | 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 wcel 1480 cvv 2686 cdif 3068 cun 3069 wss 3071 c0 3363 csn 3527 EXMIDwem 4118 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-exmid 4119 |
This theorem is referenced by: (None) |
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