Theorem suprzclex 8754

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 7486	. . . . . 6 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 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 ) ) )
4	2, 3	supclti 6614	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
5	4	ltm1d 8305	. . 3 |- ( ph -> ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) )
6		suprzclex.ss	. . . . 5 |- ( ph -> A C_ ZZ )
7		zssre 8667	. . . . 5 |- ZZ C_ RR
8	6, 7	syl6ss 3024	. . . 4 |- ( ph -> A C_ RR )
9		peano2rem 7670	. . . . 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 8325	. . 3 |- ( ph -> ( ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) <-> E. z e. A ( sup ( A , RR , < ) - 1 ) < z ) )
12	5, 11	mpbid 145	. 2 |- ( ph -> E. z e. A ( sup ( A , RR , < ) - 1 ) < z )
13	6	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ ZZ )
14	13	sselda 3012	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. ZZ )
15	7, 14	sseldi 3010	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. RR )
16	4	adantr 270	. . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. RR )
17	16	adantr 270	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> sup ( A , RR , < ) e. RR )
18		simprl 498	. . . . . . . . . . . 12 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. A )
19	13, 18	sseldd 3013	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. ZZ )
20		zre 8664	. . . . . . . . . . 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 7539	. . . . . . . . . 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 270	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( z + 1 ) e. RR )
25	3	ad2antrr 472	. . . . . . . . 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 472	. . . . . . . . 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 )
28	25, 26, 27	suprubex 8324	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ sup ( A , RR , < ) )
29		simprr 499	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) - 1 ) < z )
30		1red 7424	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> 1 e. RR )
31	16, 30, 21	ltsubaddd 7936	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( ( sup ( A , RR , < ) - 1 ) < z <-> sup ( A , RR , < ) < ( z + 1 ) ) )
32	29, 31	mpbid 145	. . . . . . . . 9 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) < ( z + 1 ) )
33	32	adantr 270	. . . . . . . 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 7529	. . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w < ( z + 1 ) )
35	19	adantr 270	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> z e. ZZ )
36		zleltp1 8715	. . . . . . . 8 |- ( ( w e. ZZ /\ z e. ZZ ) -> ( w <_ z <-> w < ( z + 1 ) ) )
37	14, 35, 36	syl2anc 403	. . . . . . 7 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> ( w <_ z <-> w < ( z + 1 ) ) )
38	34, 37	mpbird 165	. . . . . 6 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ z )
39	38	ralrimiva 2442	. . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A. w e. A w <_ z )
40		breq2 3818	. . . . . . . . . . . . 13 |- ( z = w -> ( y < z <-> y < w ) )
41	40	cbvrexv 2586	. . . . . . . . . . . 12 |- ( E. z e. A y < z <-> E. w e. A y < w )
42	41	imbi2i 224	. . . . . . . . . . 11 |- ( ( y < x -> E. z e. A y < z ) <-> ( y < x -> E. w e. A y < w ) )
43	42	ralbii 2380	. . . . . . . . . 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 445	. . . . . . . . 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 2381	. . . . . . . 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 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 ) ) )
47	46	adantr 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 ) ) )
48	13, 7	syl6ss 3024	. . . . . 6 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ RR )
49	47, 48, 21	suprleubex 8327	. . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) <_ z <-> A. w e. A w <_ z ) )
50	39, 49	mpbird 165	. . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) <_ z )
51	47, 48, 18	suprubex 8324	. . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z <_ sup ( A , RR , < ) )
52	16, 21	letri3d 7521	. . . 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 888	. . 3 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) = z )
54	53, 18	eqeltrd 2161	. 2 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. A )
55	12, 54	rexlimddv 2489	1 |- ( ph -> sup ( A , RR , < ) e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1287   e. wcel 1436   A.wral 2355   E.wrex 2356   C_ wss 2986   class class class wbr 3814  (class class class)co 5594   supcsup 6598   RRcr 7270   1c1 7272   + caddc 7274   < clt 7443   <_ cle 7444   - cmin 7574   ZZcz 8660

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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382

This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-id 4087  df-po 4090  df-iso 4091  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-iota 4937  df-fun 4974  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661

This theorem is referenced by:  infssuzcldc  10741  gcddvds  10749