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Theorem suprzclex 9280
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 7969 . . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
21adantl 275 . . . . 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 6954 . . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
54ltm1d 8818 . . 3 |- ( ph -> ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) )
6 suprzclex.ss . . . . 5 |- ( ph -> A C_ ZZ )
7 zssre 9189 . . . . 5 |- ZZ C_ RR
86, 7sstrdi 3149 . . . 4 |- ( ph -> A C_ RR )
9 peano2rem 8156 . . . . 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 8838 . . 3 |- ( ph -> ( ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) <-> E. z e. A ( sup ( A , RR , < ) - 1 ) < z ) )
125, 11mpbid 146 . 2 |- ( ph -> E. z e. A ( sup ( A , RR , < ) - 1 ) < z )
136adantr 274 . . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ ZZ )
1413sselda 3137 . . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. ZZ )
157, 14sseldi 3135 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. RR )
164adantr 274 . . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. RR )
1716adantr 274 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> sup ( A , RR , < ) e. RR )
18 simprl 521 . . . . . . . . . . . 12 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. A )
1913, 18sseldd 3138 . . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. ZZ )
20 zre 9186 . . . . . . . . . . 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 8025 . . . . . . . . . 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 274 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( z + 1 ) e. RR )
253ad2antrr 480 . . . . . . . . 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 480 . . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> A C_ RR )
27 simpr 109 . . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. A )
2825, 26, 27suprubex 8837 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ sup ( A , RR , < ) )
29 simprr 522 . . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) - 1 ) < z )
30 1red 7905 . . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> 1 e. RR )
3116, 30, 21ltsubaddd 8430 . . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( ( sup ( A , RR , < ) - 1 ) < z <-> sup ( A , RR , < ) < ( z + 1 ) ) )
3229, 31mpbid 146 . . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) < ( z + 1 ) )
3332adantr 274 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> sup ( A , RR , < ) < ( z + 1 ) )
3415, 17, 24, 28, 33lelttrd 8014 . . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w < ( z + 1 ) )
3519adantr 274 . . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> z e. ZZ )
36 zleltp1 9237 . . . . . . . 8 |- ( ( w e. ZZ /\ z e. ZZ ) -> ( w <_ z <-> w < ( z + 1 ) ) )
3714, 35, 36syl2anc 409 . . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( w <_ z <-> w < ( z + 1 ) ) )
3834, 37mpbird 166 . . . . . 6 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ z )
3938ralrimiva 2537 . . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A. w e. A w <_ z )
40 breq2 3980 . . . . . . . . . . . . 13 |- ( z = w -> ( y < z <-> y < w ) )
4140cbvrexv 2690 . . . . . . . . . . . 12 |- ( E. z e. A y < z <-> E. w e. A y < w )
4241imbi2i 225 . . . . . . . . . . 11 |- ( ( y < x -> E. z e. A y < z ) <-> ( y < x -> E. w e. A y < w ) )
4342ralbii 2470 . . . . . . . . . 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 453 . . . . . . . . 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 2471 . . . . . . . 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 121 . . . . . . 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 274 . . . . . 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, 7sstrdi 3149 . . . . . 6 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ RR )
4947, 48, 21suprleubex 8840 . . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) <_ z <-> A. w e. A w <_ z ) )
5039, 49mpbird 166 . . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) <_ z )
5147, 48, 18suprubex 8837 . . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z <_ sup ( A , RR , < ) )
5216, 21letri3d 8005 . . . 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 933 . . 3 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) = z )
5453, 18eqeltrd 2241 . 2 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. A )
5512, 54rexlimddv 2586 1 |- ( ph -> sup ( A , RR , < ) e. A )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   A.wral 2442   E.wrex 2443   C_ wss 3111   class class class wbr 3976  (class class class)co 5836   supcsup 6938   RRcr 7743   1c1 7745   + caddc 7747   < clt 7924   <_ cle 7925   - cmin 8060   ZZcz 9182
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860
This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-id 4265  df-po 4268  df-iso 4269  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-sup 6940  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183
This theorem is referenced by:  infssuzcldc  11869  gcddvds  11881
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