Theorem suprzclex 9723

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 8395	. . . . . 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 7328	. . . 4 |- ( ph -> sup ( A , RR , < ) e. RR )
5	4	ltm1d 9252	. . 3 |- ( ph -> ( sup ( A , RR , < ) - 1 ) < sup ( A , RR , < ) )
6		suprzclex.ss	. . . . 5 |- ( ph -> A C_ ZZ )
7		zssre 9630	. . . . 5 |- ZZ C_ RR
8	6, 7	sstrdi 3260	. . . 4 |- ( ph -> A C_ RR )
9		peano2rem 8583	. . . . 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 9272	. . 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 3248	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w e. ZZ )
15	7, 14	sselid 3246	. . . . . . . 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 535	. . . . . . . . . . . 12 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. A )
19	13, 18	sseldd 3249	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z e. ZZ )
20		zre 9627	. . . . . . . . . . 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 8452	. . . . . . . . . 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 492	. . . . . . . . 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 492	. . . . . . . . 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 9271	. . . . . . . 8 |- ( ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) /\ w e. A ) -> w <_ sup ( A , RR , < ) )
29		simprr 537	. . . . . . . . . 10 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> ( sup ( A , RR , < ) - 1 ) < z )
30		1red 8331	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> 1 e. RR )
31	16, 30, 21	ltsubaddd 8859	. . . . . . . . . 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 8441	. . . . . . 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 9679	. . . . . . . 8 |- ( ( w e. ZZ /\ z e. ZZ ) -> ( w <_ z <-> w < ( z + 1 ) ) )
37	14, 35, 36	syl2anc 415	. . . . . . 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 2623	. . . . 5 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A. w e. A w <_ z )
40		breq2 4129	. . . . . . . . . . . . 13 |- ( z = w -> ( y < z <-> y < w ) )
41	40	cbvrexv 2787	. . . . . . . . . . . 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 2556	. . . . . . . . . 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 461	. . . . . . . . 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 2557	. . . . . . . 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 3260	. . . . . 6 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> A C_ RR )
49	47, 48, 21	suprleubex 9274	. . . . 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 9271	. . . 4 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> z <_ sup ( A , RR , < ) )
52	16, 21	letri3d 8431	. . . 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 957	. . 3 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) = z )
54	53, 18	eqeltrd 2315	. 2 |- ( ( ph /\ ( z e. A /\ ( sup ( A , RR , < ) - 1 ) < z ) ) -> sup ( A , RR , < ) e. A )
55	12, 54	rexlimddv 2673	1 |- ( ph -> sup ( A , RR , < ) e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   class class class wbr 4125  (class class class)co 6075   supcsup 7312   RRcr 8168   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   ZZcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624

This theorem is referenced by:  infssuzcldc  10646  gcddvds  12718