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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 7837) 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 2424 | . . . . . . . . 9 | |
7 | breq1 3927 | . . . . . . . . . . . . 13 | |
8 | breq1 3927 | . . . . . . . . . . . . . 14 | |
9 | 8 | rexbidv 2436 | . . . . . . . . . . . . 13 |
10 | 7, 9 | imbi12d 233 | . . . . . . . . . . . 12 |
11 | 10 | rspcva 2782 | . . . . . . . . . . 11 |
12 | breq2 3928 | . . . . . . . . . . . 12 | |
13 | 12 | cbvrexv 2653 | . . . . . . . . . . 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 2424 | . . . . . . . . 9 | |
20 | breq1 3927 | . . . . . . . . . . . . 13 | |
21 | breq1 3927 | . . . . . . . . . . . . . 14 | |
22 | 21 | rexbidv 2436 | . . . . . . . . . . . . 13 |
23 | 20, 22 | imbi12d 233 | . . . . . . . . . . . 12 |
24 | 23 | rspcva 2782 | . . . . . . . . . . 11 |
25 | breq2 3928 | . . . . . . . . . . . 12 | |
26 | 25 | cbvrexv 2653 | . . . . . . . . . . 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 2512 | . . . . 5 |
36 | equequ1 1688 | . . . . . . 7 | |
37 | breq1 3927 | . . . . . . . . 9 | |
38 | 37 | notbid 656 | . . . . . . . 8 |
39 | breq2 3928 | . . . . . . . . 9 | |
40 | 39 | notbid 656 | . . . . . . . 8 |
41 | 38, 40 | anbi12d 464 | . . . . . . 7 |
42 | 36, 41 | bibi12d 234 | . . . . . 6 |
43 | equequ2 1689 | . . . . . . 7 | |
44 | breq2 3928 | . . . . . . . . 9 | |
45 | 44 | notbid 656 | . . . . . . . 8 |
46 | breq1 3927 | . . . . . . . . 9 | |
47 | 46 | notbid 656 | . . . . . . . 8 |
48 | 45, 47 | anbi12d 464 | . . . . . . 7 |
49 | 43, 48 | bibi12d 234 | . . . . . 6 |
50 | 42, 49 | rspc2v 2797 | . . . . 5 |
51 | 35, 50 | mpan9 279 | . . . 4 |
52 | 33, 51 | sylibrd 168 | . . 3 |
53 | 52 | ralrimivva 2512 | . 2 |
54 | breq1 3927 | . . . . . 6 | |
55 | 54 | notbid 656 | . . . . 5 |
56 | 55 | ralbidv 2435 | . . . 4 |
57 | breq2 3928 | . . . . . 6 | |
58 | 57 | imbi1d 230 | . . . . 5 |
59 | 58 | ralbidv 2435 | . . . 4 |
60 | 56, 59 | anbi12d 464 | . . 3 |
61 | 60 | rmo4 2872 | . 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 2414 wrex 2415 wrmo 2417 class class class wbr 3924 |
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 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rmo 2422 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 |
This theorem is referenced by: supeuti 6874 infmoti 6908 |
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