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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 109 | . . . 4 | |
2 | simpl3 986 | . . . 4 | |
3 | 1, 2 | sseldd 3093 | . . 3 |
4 | neldifsnd 3649 | . . 3 | |
5 | 3, 4 | pm2.65da 650 | . 2 |
6 | simplr 519 | . . . . . 6 | |
7 | simplr 519 | . . . . . . . . . . 11 | |
8 | 7 | ad2antrr 479 | . . . . . . . . . 10 |
9 | simpr 109 | . . . . . . . . . 10 | |
10 | 8, 9 | breqtrrd 3951 | . . . . . . . . 9 |
11 | breq1 3927 | . . . . . . . . . . 11 | |
12 | eleq1 2200 | . . . . . . . . . . 11 | |
13 | 11, 12 | imbi12d 233 | . . . . . . . . . 10 |
14 | simplr 519 | . . . . . . . . . 10 | |
15 | simpll3 1022 | . . . . . . . . . . 11 | |
16 | 15 | ad2antrr 479 | . . . . . . . . . 10 |
17 | 13, 14, 16 | rspcdva 2789 | . . . . . . . . 9 |
18 | 10, 17 | mpd 13 | . . . . . . . 8 |
19 | neldifsnd 3649 | . . . . . . . 8 | |
20 | 18, 19 | pm2.65da 650 | . . . . . . 7 |
21 | velsn 3539 | . . . . . . 7 | |
22 | 20, 21 | sylnibr 666 | . . . . . 6 |
23 | 6, 22 | eldifd 3076 | . . . . 5 |
24 | 23 | ex 114 | . . . 4 |
25 | 24 | ralrimiva 2503 | . . 3 |
26 | df-frind 4249 | . . . . . . . 8 FrFor | |
27 | df-frfor 4248 | . . . . . . . . 9 FrFor | |
28 | 27 | albii 1446 | . . . . . . . 8 FrFor |
29 | 26, 28 | bitri 183 | . . . . . . 7 |
30 | 29 | biimpi 119 | . . . . . 6 |
31 | 30 | 3ad2ant1 1002 | . . . . 5 |
32 | difexg 4064 | . . . . . . 7 | |
33 | eleq2 2201 | . . . . . . . . . . . . 13 | |
34 | 33 | imbi2d 229 | . . . . . . . . . . . 12 |
35 | 34 | ralbidv 2435 | . . . . . . . . . . 11 |
36 | eleq2 2201 | . . . . . . . . . . 11 | |
37 | 35, 36 | imbi12d 233 | . . . . . . . . . 10 |
38 | 37 | ralbidv 2435 | . . . . . . . . 9 |
39 | sseq2 3116 | . . . . . . . . 9 | |
40 | 38, 39 | imbi12d 233 | . . . . . . . 8 |
41 | 40 | spcgv 2768 | . . . . . . 7 |
42 | 32, 41 | syl 14 | . . . . . 6 |
43 | 42 | 3ad2ant2 1003 | . . . . 5 |
44 | 31, 43 | mpd 13 | . . . 4 |
45 | 44 | adantr 274 | . . 3 |
46 | 25, 45 | mpd 13 | . 2 |
47 | 5, 46 | mtand 654 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2414 cvv 2681 cdif 3063 wss 3066 csn 3522 class class class wbr 3924 FrFor wfrfor 4244 wfr 4245 |
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-sep 4041 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-frfor 4248 df-frind 4249 |
This theorem is referenced by: efrirr 4270 wepo 4276 wetriext 4486 |
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