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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 3979 | . . . . . 6 | |
2 | breq1 3979 | . . . . . 6 | |
3 | 1, 2 | bibi12d 234 | . . . . 5 |
4 | wetriext.ext | . . . . 5 | |
5 | wetriext.b | . . . . 5 | |
6 | 3, 4, 5 | rspcdva 2830 | . . . 4 |
7 | 6 | biimpar 295 | . . 3 |
8 | wetriext.we | . . . . . 6 | |
9 | wefr 4330 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | wetriext.a | . . . . 5 | |
12 | frirrg 4322 | . . . . 5 | |
13 | 10, 11, 5, 12 | syl3anc 1227 | . . . 4 |
14 | 13 | adantr 274 | . . 3 |
15 | 7, 14 | pm2.21dd 610 | . 2 |
16 | simpr 109 | . 2 | |
17 | breq1 3979 | . . . . . 6 | |
18 | breq1 3979 | . . . . . 6 | |
19 | 17, 18 | bibi12d 234 | . . . . 5 |
20 | wetriext.c | . . . . 5 | |
21 | 19, 4, 20 | rspcdva 2830 | . . . 4 |
22 | 21 | biimpa 294 | . . 3 |
23 | frirrg 4322 | . . . . 5 | |
24 | 10, 11, 20, 23 | syl3anc 1227 | . . . 4 |
25 | 24 | adantr 274 | . . 3 |
26 | 22, 25 | pm2.21dd 610 | . 2 |
27 | wetriext.tri | . . 3 | |
28 | breq1 3979 | . . . . . 6 | |
29 | eqeq1 2171 | . . . . . 6 | |
30 | breq2 3980 | . . . . . 6 | |
31 | 28, 29, 30 | 3orbi123d 1300 | . . . . 5 |
32 | breq2 3980 | . . . . . 6 | |
33 | eqeq2 2174 | . . . . . 6 | |
34 | breq1 3979 | . . . . . 6 | |
35 | 32, 33, 34 | 3orbi123d 1300 | . . . . 5 |
36 | 31, 35 | rspc2v 2838 | . . . 4 |
37 | 5, 20, 36 | syl2anc 409 | . . 3 |
38 | 27, 37 | mpd 13 | . 2 |
39 | 15, 16, 26, 38 | mpjao3dan 1296 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 966 wceq 1342 wcel 2135 wral 2442 class class class wbr 3976 wfr 4300 wwe 4302 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-frfor 4303 df-frind 4304 df-wetr 4306 |
This theorem is referenced by: (None) |
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