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Theorem supmoti 6929
 Description: Any class has at most one supremum in (where is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 7940) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
Hypothesis
Ref Expression
supmoti.ti
Assertion
Ref Expression
supmoti
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,   ,,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem supmoti
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ancom 264 . . . . . . 7
21anbi2ci 455 . . . . . 6
3 an42 577 . . . . . 6
4 an42 577 . . . . . 6
52, 3, 43bitr4i 211 . . . . 5
6 ralnex 2445 . . . . . . . . 9
7 breq1 3968 . . . . . . . . . . . . 13
8 breq1 3968 . . . . . . . . . . . . . 14
98rexbidv 2458 . . . . . . . . . . . . 13
107, 9imbi12d 233 . . . . . . . . . . . 12
1110rspcva 2814 . . . . . . . . . . 11
12 breq2 3969 . . . . . . . . . . . 12
1312cbvrexv 2681 . . . . . . . . . . 11
1411, 13syl6ibr 161 . . . . . . . . . 10
1514con3d 621 . . . . . . . . 9
166, 15syl5bi 151 . . . . . . . 8
1716expimpd 361 . . . . . . 7
1817ad2antrl 482 . . . . . 6
19 ralnex 2445 . . . . . . . . 9
20 breq1 3968 . . . . . . . . . . . . 13
21 breq1 3968 . . . . . . . . . . . . . 14
2221rexbidv 2458 . . . . . . . . . . . . 13
2320, 22imbi12d 233 . . . . . . . . . . . 12
2423rspcva 2814 . . . . . . . . . . 11
25 breq2 3969 . . . . . . . . . . . 12
2625cbvrexv 2681 . . . . . . . . . . 11
2724, 26syl6ibr 161 . . . . . . . . . 10
2827con3d 621 . . . . . . . . 9
2919, 28syl5bi 151 . . . . . . . 8
3029expimpd 361 . . . . . . 7
3130ad2antll 483 . . . . . 6
3218, 31anim12d 333 . . . . 5
335, 32syl5bi 151 . . . 4
34 supmoti.ti . . . . . 6
3534ralrimivva 2539 . . . . 5
36 equequ1 1692 . . . . . . 7
37 breq1 3968 . . . . . . . . 9
3837notbid 657 . . . . . . . 8
39 breq2 3969 . . . . . . . . 9
4039notbid 657 . . . . . . . 8
4138, 40anbi12d 465 . . . . . . 7
4236, 41bibi12d 234 . . . . . 6
43 equequ2 1693 . . . . . . 7
44 breq2 3969 . . . . . . . . 9
4544notbid 657 . . . . . . . 8
46 breq1 3968 . . . . . . . . 9
4746notbid 657 . . . . . . . 8
4845, 47anbi12d 465 . . . . . . 7
4943, 48bibi12d 234 . . . . . 6
5042, 49rspc2v 2829 . . . . 5
5135, 50mpan9 279 . . . 4
5233, 51sylibrd 168 . . 3
5352ralrimivva 2539 . 2
54 breq1 3968 . . . . . 6
5554notbid 657 . . . . 5
5655ralbidv 2457 . . . 4
57 breq2 3969 . . . . . 6
5857imbi1d 230 . . . . 5
5958ralbidv 2457 . . . 4
6056, 59anbi12d 465 . . 3
6160rmo4 2905 . 2
6253, 61sylibr 133 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wcel 2128  wral 2435  wrex 2436  wrmo 2438   class class class wbr 3965 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rmo 2443  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966 This theorem is referenced by:  supeuti  6930  infmoti  6964
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