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Mirrors > Home > ILE Home > Th. List > wetriext | Unicode version |
Description: A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
Ref | Expression |
---|---|
wetriext.we | |
wetriext.a | |
wetriext.tri | |
wetriext.b | |
wetriext.c | |
wetriext.ext |
Ref | Expression |
---|---|
wetriext |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3932 | . . . . . 6 | |
2 | breq1 3932 | . . . . . 6 | |
3 | 1, 2 | bibi12d 234 | . . . . 5 |
4 | wetriext.ext | . . . . 5 | |
5 | wetriext.b | . . . . 5 | |
6 | 3, 4, 5 | rspcdva 2794 | . . . 4 |
7 | 6 | biimpar 295 | . . 3 |
8 | wetriext.we | . . . . . 6 | |
9 | wefr 4280 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | wetriext.a | . . . . 5 | |
12 | frirrg 4272 | . . . . 5 | |
13 | 10, 11, 5, 12 | syl3anc 1216 | . . . 4 |
14 | 13 | adantr 274 | . . 3 |
15 | 7, 14 | pm2.21dd 609 | . 2 |
16 | simpr 109 | . 2 | |
17 | breq1 3932 | . . . . . 6 | |
18 | breq1 3932 | . . . . . 6 | |
19 | 17, 18 | bibi12d 234 | . . . . 5 |
20 | wetriext.c | . . . . 5 | |
21 | 19, 4, 20 | rspcdva 2794 | . . . 4 |
22 | 21 | biimpa 294 | . . 3 |
23 | frirrg 4272 | . . . . 5 | |
24 | 10, 11, 20, 23 | syl3anc 1216 | . . . 4 |
25 | 24 | adantr 274 | . . 3 |
26 | 22, 25 | pm2.21dd 609 | . 2 |
27 | wetriext.tri | . . 3 | |
28 | breq1 3932 | . . . . . 6 | |
29 | eqeq1 2146 | . . . . . 6 | |
30 | breq2 3933 | . . . . . 6 | |
31 | 28, 29, 30 | 3orbi123d 1289 | . . . . 5 |
32 | breq2 3933 | . . . . . 6 | |
33 | eqeq2 2149 | . . . . . 6 | |
34 | breq1 3932 | . . . . . 6 | |
35 | 32, 33, 34 | 3orbi123d 1289 | . . . . 5 |
36 | 31, 35 | rspc2v 2802 | . . . 4 |
37 | 5, 20, 36 | syl2anc 408 | . . 3 |
38 | 27, 37 | mpd 13 | . 2 |
39 | 15, 16, 26, 38 | mpjao3dan 1285 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 961 wceq 1331 wcel 1480 wral 2416 class class class wbr 3929 wfr 4250 wwe 4252 |
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 2121 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-frfor 4253 df-frind 4254 df-wetr 4256 |
This theorem is referenced by: (None) |
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