Theorem suplocexpr 7674

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 3990	. . . . . 6 |- ( a = w -> ( a <Q u <-> w <Q u ) )
5	4	cbvrexv 2697	. . . . 5 |- ( E. a e. |^| ( 2nd " A ) a <Q u <-> E. w e. |^| ( 2nd " A ) w <Q u )
6	5	rabbii 2716	. . . 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 3767	. . 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 7671	. 2 |- ( ph -> <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. e. P. )
9	1, 2, 3, 7	suplocexprlemub 7672	. 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 7673	. . 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 2544	. 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 3990	. . . . . 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 662	. . . . 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 2470	. . . 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 3991	. . . . . 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 230	. . . . 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 2470	. . . 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 470	. . 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 2834	. 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 1231	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 103   \/ wo 703   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3584   U.cuni 3794   |^|cint 3829   class class class wbr 3987   "cima 4612   1stc1st 6114   2ndc2nd 6115   Q.cnq 7229   <Q cltq 7234   P.cnp 7240   <P cltp 7244

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-2o 6393  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-enq0 7373  df-nq0 7374  df-0nq0 7375  df-plq0 7376  df-mq0 7377  df-inp 7415  df-iltp 7419

This theorem is referenced by:  suplocsrlempr  7756