Theorem suplocexpr 7935

Description: An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	|- ( ph -> E. x x e. A )
suplocexpr.ub	|- ( ph -> E. x e. P. A. y e. A y <P x )
suplocexpr.loc	|- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )

Assertion

Ref	Expression
suplocexpr	|- ( ph -> E. x e. P. ( A. y e. A -. x <P y /\ A. y e. P. ( y <P x -> E. z e. A y <P z ) ) )

Distinct variable groups:   y, A, z, x   ph, y, z, x

Proof of Theorem suplocexpr

Dummy variables a u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . 3 |- ( ph -> E. x x e. A )
2		suplocexpr.ub	. . 3 |- ( ph -> E. x e. P. A. y e. A y <P x )
3		suplocexpr.loc	. . 3 |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )
4		breq1 4089	. . . . . 6 |- ( a = w -> ( a <Q u <-> w <Q u ) )
5	4	cbvrexv 2766	. . . . 5 |- ( E. a e. |^| ( 2nd " A ) a <Q u <-> E. w e. |^| ( 2nd " A ) w <Q u )
6	5	rabbii 2787	. . . 4 |- { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u }
7	6	opeq2i 3864	. . 3 |- <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
8	1, 2, 3, 7	suplocexprlemex 7932	. 2 |- ( ph -> <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. e. P. )
9	1, 2, 3, 7	suplocexprlemub 7933	. 2 |- ( ph -> A. y e. A -. <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. <P y )
10	1, 2, 3, 7	suplocexprlemlub 7934	. . 3 |- ( ph -> ( y <P <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> E. z e. A y <P z ) )
11	10	ralrimivw 2604	. 2 |- ( ph -> A. y e. P. ( y <P <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> E. z e. A y <P z ) )
12		breq1 4089	. . . . . 6 |- ( x = <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> ( x <P y <-> <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. <P y ) )
13	12	notbid 671	. . . . 5 |- ( x = <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> ( -. x <P y <-> -. <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. <P y ) )
14	13	ralbidv 2530	. . . 4 |- ( x = <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> ( A. y e. A -. x <P y <-> A. y e. A -. <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. <P y ) )
15		breq2 4090	. . . . . 6 |- ( x = <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> ( y <P x <-> y <P <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. ) )
16	15	imbi1d 231	. . . . 5 |- ( x = <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> ( ( y <P x -> E. z e. A y <P z ) <-> ( y <P <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> E. z e. A y <P z ) ) )
17	16	ralbidv 2530	. . . 4 |- ( x = <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> ( A. y e. P. ( y <P x -> E. z e. A y <P z ) <-> A. y e. P. ( y <P <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> E. z e. A y <P z ) ) )
18	14, 17	anbi12d 473	. . 3 |- ( x = <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> ( ( A. y e. A -. x <P y /\ A. y e. P. ( y <P x -> E. z e. A y <P z ) ) <-> ( A. y e. A -. <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. <P y /\ A. y e. P. ( y <P <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> E. z e. A y <P z ) ) ) )
19	18	rspcev 2908	. 2 |- ( ( <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. e. P. /\ ( A. y e. A -. <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. <P y /\ A. y e. P. ( y <P <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. -> E. z e. A y <P z ) ) ) -> E. x e. P. ( A. y e. A -. x <P y /\ A. y e. P. ( y <P x -> E. z e. A y <P z ) ) )
20	8, 9, 11, 19	syl12anc 1269	1 |- ( ph -> E. x e. P. ( A. y e. A -. x <P y /\ A. y e. P. ( y <P x -> E. z e. A y <P z ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3670   U.cuni 3891   |^|cint 3926   class class class wbr 4086   "cima 4726   1stc1st 6296   2ndc2nd 6297   Q.cnq 7490   <Q cltq 7495   P.cnp 7501   <P cltp 7505

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-iltp 7680

This theorem is referenced by:  suplocsrlempr  8017