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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 103 | . . . 4 | |
2 | simpl3 909 | . . . 4 | |
3 | 1, 2 | sseldd 2946 | . . 3 |
4 | neldifsnd 3498 | . . 3 | |
5 | 3, 4 | pm2.65da 587 | . 2 |
6 | simplr 482 | . . . . . 6 | |
7 | simpll3 945 | . . . . . . . . . 10 | |
8 | 7 | ad2antrr 457 | . . . . . . . . 9 |
9 | simplr 482 | . . . . . . . . 9 | |
10 | simplr 482 | . . . . . . . . . . 11 | |
11 | 10 | ad2antrr 457 | . . . . . . . . . 10 |
12 | simpr 103 | . . . . . . . . . 10 | |
13 | 11, 12 | breqtrrd 3790 | . . . . . . . . 9 |
14 | breq1 3767 | . . . . . . . . . . 11 | |
15 | eleq1 2100 | . . . . . . . . . . 11 | |
16 | 14, 15 | imbi12d 223 | . . . . . . . . . 10 |
17 | 16 | rspcv 2652 | . . . . . . . . 9 |
18 | 8, 9, 13, 17 | syl3c 57 | . . . . . . . 8 |
19 | neldifsnd 3498 | . . . . . . . 8 | |
20 | 18, 19 | pm2.65da 587 | . . . . . . 7 |
21 | velsn 3392 | . . . . . . 7 | |
22 | 20, 21 | sylnibr 602 | . . . . . 6 |
23 | 6, 22 | eldifd 2928 | . . . . 5 |
24 | 23 | ex 108 | . . . 4 |
25 | 24 | ralrimiva 2392 | . . 3 |
26 | df-frind 4069 | . . . . . . . 8 FrFor | |
27 | df-frfor 4068 | . . . . . . . . 9 FrFor | |
28 | 27 | albii 1359 | . . . . . . . 8 FrFor |
29 | 26, 28 | bitri 173 | . . . . . . 7 |
30 | 29 | biimpi 113 | . . . . . 6 |
31 | 30 | 3ad2ant1 925 | . . . . 5 |
32 | difexg 3898 | . . . . . . 7 | |
33 | eleq2 2101 | . . . . . . . . . . . . 13 | |
34 | 33 | imbi2d 219 | . . . . . . . . . . . 12 |
35 | 34 | ralbidv 2326 | . . . . . . . . . . 11 |
36 | eleq2 2101 | . . . . . . . . . . 11 | |
37 | 35, 36 | imbi12d 223 | . . . . . . . . . 10 |
38 | 37 | ralbidv 2326 | . . . . . . . . 9 |
39 | sseq2 2967 | . . . . . . . . 9 | |
40 | 38, 39 | imbi12d 223 | . . . . . . . 8 |
41 | 40 | spcgv 2640 | . . . . . . 7 |
42 | 32, 41 | syl 14 | . . . . . 6 |
43 | 42 | 3ad2ant2 926 | . . . . 5 |
44 | 31, 43 | mpd 13 | . . . 4 |
45 | 44 | adantr 261 | . . 3 |
46 | 25, 45 | mpd 13 | . 2 |
47 | 5, 46 | mtand 591 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 w3a 885 wal 1241 wceq 1243 wcel 1393 wral 2306 cvv 2557 cdif 2914 wss 2917 csn 3375 class class class wbr 3764 FrFor wfrfor 4064 wfr 4065 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-frfor 4068 df-frind 4069 |
This theorem is referenced by: efrirr 4090 wepo 4096 wetriext 4301 |
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