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Mirrors > Home > ILE Home > Th. List > maxleim | Unicode version |
Description: Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.) |
Ref | Expression |
---|---|
maxleim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7191 | . . . 4 | |
2 | 1 | adantl 271 | . . 3 |
3 | simplr 496 | . . 3 | |
4 | prid2g 3497 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | simpll 495 | . . . . . . 7 | |
7 | 6 | ad2antrr 471 | . . . . . 6 |
8 | 3 | ad2antrr 471 | . . . . . 6 |
9 | simpllr 500 | . . . . . 6 | |
10 | 7, 8, 9 | lensymd 7231 | . . . . 5 |
11 | breq2 3789 | . . . . . . 7 | |
12 | 11 | notbid 624 | . . . . . 6 |
13 | 12 | adantl 271 | . . . . 5 |
14 | 10, 13 | mpbird 165 | . . . 4 |
15 | 3 | ad2antrr 471 | . . . . . 6 |
16 | 15 | ltnrd 7222 | . . . . 5 |
17 | breq2 3789 | . . . . . . 7 | |
18 | 17 | notbid 624 | . . . . . 6 |
19 | 18 | adantl 271 | . . . . 5 |
20 | 16, 19 | mpbird 165 | . . . 4 |
21 | elpri 3421 | . . . . 5 | |
22 | 21 | adantl 271 | . . . 4 |
23 | 14, 20, 22 | mpjaodan 744 | . . 3 |
24 | 2, 3, 5, 23 | supmaxti 6417 | . 2 |
25 | 24 | ex 113 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 cpr 3399 class class class wbr 3785 csup 6395 cr 6980 clt 7153 cle 7154 |
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-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-apti 7091 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-iota 4887 df-riota 5488 df-sup 6397 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 |
This theorem is referenced by: maxleb 10102 |
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