Theorem suplocexpr 8040

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 4112	. . . . . 6 |- ( a = w -> ( a <Q u <-> w <Q u ) )
5	4	cbvrexv 2779	. . . . 5 |- ( E. a e. |^| ( 2nd " A ) a <Q u <-> E. w e. |^| ( 2nd " A ) w <Q u )
6	5	rabbii 2800	. . . 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 3887	. . 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 8037	. 2 |- ( ph -> <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. e. P. )
9	1, 2, 3, 7	suplocexprlemub 8038	. 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 8039	. . 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 2616	. 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 4112	. . . . . 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 673	. . . . 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 2542	. . . 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 4113	. . . . . 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 2542	. . . 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 2921	. 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 1272	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 716   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   <.cop 3692   U.cuni 3914   |^|cint 3949   class class class wbr 4109   "cima 4752   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   <Q cltq 7600   P.cnp 7606   <P cltp 7610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iltp 7785

This theorem is referenced by:  suplocsrlempr  8122