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Theorem suprzclex 8445
Description: The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
Hypotheses
Ref Expression
suprzclex.ex |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
suprzclex.ss |- ( ph -> A C_ ZZ )
Assertion
Ref Expression
suprzclex |- ( ph -> sup ( A , RR , < ) e. A )
Distinct variable groups:   x, A, y, z   ph, x, z
Allowed substitution hint:   ph( y)
Proof of Theorem suprzclex
Dummy variables w f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lttri3 7191 . . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
21adantl 271 . . . . 5 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3 suprzclex.ex . . . . 5 |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
42, 3supclti 6411 . . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
54ltm1d 8010 . . 3 |- ( ph -> ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) )
6 suprzclex.ss . . . . 5 |- ( ph -> A C_ ZZ )
7 zssre 8358 . . . . 5 |- ZZ C_ RR
86, 7syl6ss 3011 . . . 4 |- ( ph -> A C_ RR )
9 peano2rem 7375 . . . . 5 |- ( sup ( A , RR , < ) e. RR -> ( sup ( A , RR , < ) - 1 ) e. RR )
104, 9syl 14 . . . 4 |- ( ph -> ( sup ( A , RR , < ) - 1 ) e. RR )
113, 8, 10suprlubex 8030 . . 3 |- ( ph -> ( ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) <-> E. z e. A ( sup ( A , RR , < ) - 1 ) < z ) )
125, 11mpbid 145 . 2 |- ( ph -> E. z e. A ( sup ( A , RR , < ) - 1 ) < z )
136adantr 270 . . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ ZZ )
1413sselda 2999 . . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. ZZ )
157, 14sseldi 2997 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. RR )
164adantr 270 . . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. RR )
1716adantr 270 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> sup ( A , RR , < ) e. RR )
18 simprl 497 . . . . . . . . . . . 12 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. A )
1913, 18sseldd 3000 . . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. ZZ )
20 zre 8355 . . . . . . . . . . 11 |- ( z e. ZZ -> z e. RR )
2119, 20syl 14 . . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. RR )
22 peano2re 7244 . . . . . . . . . 10 |- ( z e. RR -> ( z + 1 ) e. RR )
2321, 22syl 14 . . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( z + 1 ) e. RR )
2423adantr 270 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( z + 1 ) e. RR )
253ad2antrr 471 . . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
268ad2antrr 471 . . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> A C_ RR )
27 simpr 108 . . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. A )
2825, 26, 27suprubex 8029 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ sup ( A , RR , < ) )
29 simprr 498 . . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) - 1 ) < z )
30 1red 7134 . . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> 1 e. RR )
3116, 30, 21ltsubaddd 7641 . . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( ( sup ( A , RR , < ) - 1 ) < z <-> sup ( A , RR , < ) < ( z + 1 ) ) )
3229, 31mpbid 145 . . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) < ( z + 1 ) )
3332adantr 270 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> sup ( A , RR , < ) < ( z + 1 ) )
3415, 17, 24, 28, 33lelttrd 7234 . . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w < ( z + 1 ) )
3519adantr 270 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> z e. ZZ )
36 zleltp1 8406 . . . . . . . 8 |- ( ( w e. ZZ /\ z e. ZZ ) -> ( w <_ z <-> w < ( z + 1 ) ) )
3714, 35, 36syl2anc 403 . . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( w <_ z <-> w < ( z + 1 ) ) )
3834, 37mpbird 165 . . . . . 6 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ z )
3938ralrimiva 2434 . . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A. w e. A w <_ z )
40 breq2 3789 . . . . . . . . . . . . 13 |- ( z = w -> ( y < z <-> y < w ) )
4140cbvrexv 2578 . . . . . . . . . . . 12 |- ( E. z e. A y < z <-> E. w e. A y < w )
4241imbi2i 224 . . . . . . . . . . 11 |- ( ( y < x -> E. z e. A y < z ) <-> ( y < x -> E. w e. A y < w ) )
4342ralbii 2372 . . . . . . . . . 10 |- ( A. y e. RR ( y < x -> E. z e. A y < z ) <-> A. y e. RR ( y < x -> E. w e. A y < w ) )
4443anbi2i 444 . . . . . . . . 9 |- ( ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) <-> ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. w e. A y < w ) ) )
4544rexbii 2373 . . . . . . . 8 |- ( E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) <-> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. w e. A y < w ) ) )
463, 45sylib 120 . . . . . . 7 |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. w e. A y < w ) ) )
4746adantr 270 . . . . . 6 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. w e. A y < w ) ) )
4813, 7syl6ss 3011 . . . . . 6 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ RR )
4947, 48, 21suprleubex 8032 . . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) <_ z <-> A. w e. A w <_ z ) )
5039, 49mpbird 165 . . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) <_ z )
5147, 48, 18suprubex 8029 . . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z <_ sup ( A , RR , < ) )
5216, 21letri3d 7226 . . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) = z <-> ( sup ( A , RR , < ) <_ z /\ z <_ sup ( A , RR , < ) ) ) )
5350, 51, 52mpbir2and 885 . . 3 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) = z )
5453, 18eqeltrd 2155 . 2 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. A )
5512, 54rexlimddv 2481 1 |- ( ph -> sup ( A , RR , < ) e. A )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   C_ wss 2973   class class class wbr 3785  (class class class)co 5532   supcsup 6395   RRcr 6980   1c1 6982   + caddc 6984   < clt 7153   <_ cle 7154   - cmin 7279   ZZcz 8351
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-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092
This theorem depends on definitions:  df-bi 115  df-3or 920  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-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-sup 6397  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352
This theorem is referenced by:  infssuzcldc  10347  gcddvds  10355
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