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Mirrors > Home > ILE Home > Th. List > supmoti | Unicode version |
Description: Any class has at most one supremum in (where is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 7847) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.) |
Ref | Expression |
---|---|
supmoti.ti |
Ref | Expression |
---|---|
supmoti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . . . . . . 7 | |
2 | 1 | anbi2ci 454 | . . . . . 6 |
3 | an42 576 | . . . . . 6 | |
4 | an42 576 | . . . . . 6 | |
5 | 2, 3, 4 | 3bitr4i 211 | . . . . 5 |
6 | ralnex 2426 | . . . . . . . . 9 | |
7 | breq1 3932 | . . . . . . . . . . . . 13 | |
8 | breq1 3932 | . . . . . . . . . . . . . 14 | |
9 | 8 | rexbidv 2438 | . . . . . . . . . . . . 13 |
10 | 7, 9 | imbi12d 233 | . . . . . . . . . . . 12 |
11 | 10 | rspcva 2787 | . . . . . . . . . . 11 |
12 | breq2 3933 | . . . . . . . . . . . 12 | |
13 | 12 | cbvrexv 2655 | . . . . . . . . . . 11 |
14 | 11, 13 | syl6ibr 161 | . . . . . . . . . 10 |
15 | 14 | con3d 620 | . . . . . . . . 9 |
16 | 6, 15 | syl5bi 151 | . . . . . . . 8 |
17 | 16 | expimpd 360 | . . . . . . 7 |
18 | 17 | ad2antrl 481 | . . . . . 6 |
19 | ralnex 2426 | . . . . . . . . 9 | |
20 | breq1 3932 | . . . . . . . . . . . . 13 | |
21 | breq1 3932 | . . . . . . . . . . . . . 14 | |
22 | 21 | rexbidv 2438 | . . . . . . . . . . . . 13 |
23 | 20, 22 | imbi12d 233 | . . . . . . . . . . . 12 |
24 | 23 | rspcva 2787 | . . . . . . . . . . 11 |
25 | breq2 3933 | . . . . . . . . . . . 12 | |
26 | 25 | cbvrexv 2655 | . . . . . . . . . . 11 |
27 | 24, 26 | syl6ibr 161 | . . . . . . . . . 10 |
28 | 27 | con3d 620 | . . . . . . . . 9 |
29 | 19, 28 | syl5bi 151 | . . . . . . . 8 |
30 | 29 | expimpd 360 | . . . . . . 7 |
31 | 30 | ad2antll 482 | . . . . . 6 |
32 | 18, 31 | anim12d 333 | . . . . 5 |
33 | 5, 32 | syl5bi 151 | . . . 4 |
34 | supmoti.ti | . . . . . 6 | |
35 | 34 | ralrimivva 2514 | . . . . 5 |
36 | equequ1 1688 | . . . . . . 7 | |
37 | breq1 3932 | . . . . . . . . 9 | |
38 | 37 | notbid 656 | . . . . . . . 8 |
39 | breq2 3933 | . . . . . . . . 9 | |
40 | 39 | notbid 656 | . . . . . . . 8 |
41 | 38, 40 | anbi12d 464 | . . . . . . 7 |
42 | 36, 41 | bibi12d 234 | . . . . . 6 |
43 | equequ2 1689 | . . . . . . 7 | |
44 | breq2 3933 | . . . . . . . . 9 | |
45 | 44 | notbid 656 | . . . . . . . 8 |
46 | breq1 3932 | . . . . . . . . 9 | |
47 | 46 | notbid 656 | . . . . . . . 8 |
48 | 45, 47 | anbi12d 464 | . . . . . . 7 |
49 | 43, 48 | bibi12d 234 | . . . . . 6 |
50 | 42, 49 | rspc2v 2802 | . . . . 5 |
51 | 35, 50 | mpan9 279 | . . . 4 |
52 | 33, 51 | sylibrd 168 | . . 3 |
53 | 52 | ralrimivva 2514 | . 2 |
54 | breq1 3932 | . . . . . 6 | |
55 | 54 | notbid 656 | . . . . 5 |
56 | 55 | ralbidv 2437 | . . . 4 |
57 | breq2 3933 | . . . . . 6 | |
58 | 57 | imbi1d 230 | . . . . 5 |
59 | 58 | ralbidv 2437 | . . . 4 |
60 | 56, 59 | anbi12d 464 | . . 3 |
61 | 60 | rmo4 2877 | . 2 |
62 | 53, 61 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 1480 wral 2416 wrex 2417 wrmo 2419 class class class wbr 3929 |
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 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rmo 2424 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 |
This theorem is referenced by: supeuti 6881 infmoti 6915 |
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