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Theorem toslub 28108
 Description: In a toset, the lowest upper bound , defined for partial orders is the supremum, , defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)
Hypotheses
Ref Expression
toslub.b
toslub.l
toslub.1 Toset
toslub.2
Assertion
Ref Expression
toslub

Proof of Theorem toslub
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toslub.b . . . 4
2 toslub.l . . . 4
3 toslub.1 . . . 4 Toset
4 toslub.2 . . . 4
5 eqid 2402 . . . 4
61, 2, 3, 4, 5toslublem 28107 . . 3
76riotabidva 6256 . 2
8 eqid 2402 . . 3
9 biid 236 . . 3
101, 5, 8, 9, 3, 4lubval 15938 . 2
111, 5, 2tosso 15990 . . . . 5 Toset Toset
1211ibi 241 . . . 4 Toset
1312simpld 457 . . 3 Toset
14 id 22 . . . 4
1514supval2 7948 . . 3
163, 13, 153syl 18 . 2
177, 10, 163eqtr4d 2453 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 367   wceq 1405   wcel 1842  wral 2754  wrex 2755   wss 3414   class class class wbr 4395   cid 4733   wor 4743   cres 4825  cfv 5569  crio 6239  csup 7934  cbs 14841  cple 14916  cplt 15894  club 15895  Tosetctos 15987 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-sup 7935  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-toset 15988 This theorem is referenced by:  xrsp1  28122
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