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Mirrors > Home > ILE Home > Th. List > frirrg | Unicode version |
Description: A well-founded relation is irreflexive. This is the case where exists. (Contributed by Jim Kingdon, 21-Sep-2021.) |
Ref | Expression |
---|---|
frirrg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 108 | . . . 4 | |
2 | simpl3 943 | . . . 4 | |
3 | 1, 2 | sseldd 3000 | . . 3 |
4 | neldifsnd 3520 | . . 3 | |
5 | 3, 4 | pm2.65da 619 | . 2 |
6 | simplr 496 | . . . . . 6 | |
7 | simpll3 979 | . . . . . . . . . 10 | |
8 | 7 | ad2antrr 471 | . . . . . . . . 9 |
9 | simplr 496 | . . . . . . . . 9 | |
10 | simplr 496 | . . . . . . . . . . 11 | |
11 | 10 | ad2antrr 471 | . . . . . . . . . 10 |
12 | simpr 108 | . . . . . . . . . 10 | |
13 | 11, 12 | breqtrrd 3811 | . . . . . . . . 9 |
14 | breq1 3788 | . . . . . . . . . . 11 | |
15 | eleq1 2141 | . . . . . . . . . . 11 | |
16 | 14, 15 | imbi12d 232 | . . . . . . . . . 10 |
17 | 16 | rspcv 2697 | . . . . . . . . 9 |
18 | 8, 9, 13, 17 | syl3c 62 | . . . . . . . 8 |
19 | neldifsnd 3520 | . . . . . . . 8 | |
20 | 18, 19 | pm2.65da 619 | . . . . . . 7 |
21 | velsn 3415 | . . . . . . 7 | |
22 | 20, 21 | sylnibr 634 | . . . . . 6 |
23 | 6, 22 | eldifd 2983 | . . . . 5 |
24 | 23 | ex 113 | . . . 4 |
25 | 24 | ralrimiva 2434 | . . 3 |
26 | df-frind 4087 | . . . . . . . 8 FrFor | |
27 | df-frfor 4086 | . . . . . . . . 9 FrFor | |
28 | 27 | albii 1399 | . . . . . . . 8 FrFor |
29 | 26, 28 | bitri 182 | . . . . . . 7 |
30 | 29 | biimpi 118 | . . . . . 6 |
31 | 30 | 3ad2ant1 959 | . . . . 5 |
32 | difexg 3919 | . . . . . . 7 | |
33 | eleq2 2142 | . . . . . . . . . . . . 13 | |
34 | 33 | imbi2d 228 | . . . . . . . . . . . 12 |
35 | 34 | ralbidv 2368 | . . . . . . . . . . 11 |
36 | eleq2 2142 | . . . . . . . . . . 11 | |
37 | 35, 36 | imbi12d 232 | . . . . . . . . . 10 |
38 | 37 | ralbidv 2368 | . . . . . . . . 9 |
39 | sseq2 3021 | . . . . . . . . 9 | |
40 | 38, 39 | imbi12d 232 | . . . . . . . 8 |
41 | 40 | spcgv 2685 | . . . . . . 7 |
42 | 32, 41 | syl 14 | . . . . . 6 |
43 | 42 | 3ad2ant2 960 | . . . . 5 |
44 | 31, 43 | mpd 13 | . . . 4 |
45 | 44 | adantr 270 | . . 3 |
46 | 25, 45 | mpd 13 | . 2 |
47 | 5, 46 | mtand 623 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 w3a 919 wal 1282 wceq 1284 wcel 1433 wral 2348 cvv 2601 cdif 2970 wss 2973 csn 3398 class class class wbr 3785 FrFor wfrfor 4082 wfr 4083 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-frfor 4086 df-frind 4087 |
This theorem is referenced by: efrirr 4108 wepo 4114 wetriext 4319 |
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