Theorem suprzclex 9577

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.

Step	Hyp	Ref	Expression
1		lttri3 8258	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . . . 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 ) ) )
4	2, 3	supclti 7196	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
5	4	ltm1d 9111	. . 3 |- ( ph -> ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) )
6		suprzclex.ss	. . . . 5 |- ( ph -> A C_ ZZ )
7		zssre 9485	. . . . 5 |- ZZ C_ RR
8	6, 7	sstrdi 3239	. . . 4 |- ( ph -> A C_ RR )
9		peano2rem 8445	. . . . 5 |- ( sup ( A , RR , < ) e. RR -> ( sup ( A , RR , < ) - 1 ) e. RR )
10	4, 9	syl 14	. . . 4 |- ( ph -> ( sup ( A , RR , < ) - 1 ) e. RR )
11	3, 8, 10	suprlubex 9131	. . 3 |- ( ph -> ( ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) <-> E. z e. A ( sup ( A , RR , < ) - 1 ) < z ) )
12	5, 11	mpbid 147	. 2 |- ( ph -> E. z e. A ( sup ( A , RR , < ) - 1 ) < z )
13	6	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ ZZ )
14	13	sselda 3227	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. ZZ )
15	7, 14	sselid 3225	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. RR )
16	4	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. RR )
17	16	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> sup ( A , RR , < ) e. RR )
18		simprl 531	. . . . . . . . . . . 12 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. A )
19	13, 18	sseldd 3228	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. ZZ )
20		zre 9482	. . . . . . . . . . 11 |- ( z e. ZZ -> z e. RR )
21	19, 20	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. RR )
22		peano2re 8314	. . . . . . . . . 10 |- ( z e. RR -> ( z + 1 ) e. RR )
23	21, 22	syl 14	. . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( z + 1 ) e. RR )
24	23	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( z + 1 ) e. RR )
25	3	ad2antrr 488	. . . . . . . . 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 ) ) )
26	8	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> A C_ RR )
27		simpr 110	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. A )
28	25, 26, 27	suprubex 9130	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ sup ( A , RR , < ) )
29		simprr 533	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) - 1 ) < z )
30		1red 8193	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> 1 e. RR )
31	16, 30, 21	ltsubaddd 8720	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( ( sup ( A , RR , < ) - 1 ) < z <-> sup ( A , RR , < ) < ( z + 1 ) ) )
32	29, 31	mpbid 147	. . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) < ( z + 1 ) )
33	32	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> sup ( A , RR , < ) < ( z + 1 ) )
34	15, 17, 24, 28, 33	lelttrd 8303	. . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w < ( z + 1 ) )
35	19	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> z e. ZZ )
36		zleltp1 9534	. . . . . . . 8 |- ( ( w e. ZZ /\ z e. ZZ ) -> ( w <_ z <-> w < ( z + 1 ) ) )
37	14, 35, 36	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( w <_ z <-> w < ( z + 1 ) ) )
38	34, 37	mpbird 167	. . . . . 6 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ z )
39	38	ralrimiva 2605	. . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A. w e. A w <_ z )
40		breq2 4092	. . . . . . . . . . . . 13 |- ( z = w -> ( y < z <-> y < w ) )
41	40	cbvrexv 2768	. . . . . . . . . . . 12 |- ( E. z e. A y < z <-> E. w e. A y < w )
42	41	imbi2i 226	. . . . . . . . . . 11 |- ( ( y < x -> E. z e. A y < z ) <-> ( y < x -> E. w e. A y < w ) )
43	42	ralbii 2538	. . . . . . . . . 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 ) )
44	43	anbi2i 457	. . . . . . . . 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 ) ) )
45	44	rexbii 2539	. . . . . . . 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 ) ) )
46	3, 45	sylib 122	. . . . . . 7 |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. w e. A y < w ) ) )
47	46	adantr 276	. . . . . 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 ) ) )
48	13, 7	sstrdi 3239	. . . . . 6 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ RR )
49	47, 48, 21	suprleubex 9133	. . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) <_ z <-> A. w e. A w <_ z ) )
50	39, 49	mpbird 167	. . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) <_ z )
51	47, 48, 18	suprubex 9130	. . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z <_ sup ( A , RR , < ) )
52	16, 21	letri3d 8294	. . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) = z <-> ( sup ( A , RR , < ) <_ z /\ z <_ sup ( A , RR , < ) ) ) )
53	50, 51, 52	mpbir2and 952	. . 3 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) = z )
54	53, 18	eqeltrd 2308	. 2 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. A )
55	12, 54	rexlimddv 2655	1 |- ( ph -> sup ( A , RR , < ) e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   C_ wss 3200   class class class wbr 4088  (class class class)co 6017   supcsup 7180   RRcr 8030   1c1 8032   + caddc 8034   < clt 8213   <_ cle 8214   - cmin 8349   ZZcz 9478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479

This theorem is referenced by:  infssuzcldc  10494  gcddvds  12533