Theorem suplocexpr 7753

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 4021	. . . . . 6 |- ( a = w -> ( a <Q u <-> w <Q u ) )
5	4	cbvrexv 2719	. . . . 5 |- ( E. a e. |^| ( 2nd " A ) a <Q u <-> E. w e. |^| ( 2nd " A ) w <Q u )
6	5	rabbii 2738	. . . 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 3797	. . 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 7750	. 2 |- ( ph -> <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. e. P. )
9	1, 2, 3, 7	suplocexprlemub 7751	. 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 7752	. . 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 2564	. 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 4021	. . . . . 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 668	. . . . 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 2490	. . . 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 4022	. . . . . 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 2490	. . . 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 2856	. 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 1247	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 709   = wceq 1364   E.wex 1503   e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   <.cop 3610   U.cuni 3824   |^|cint 3859   class class class wbr 4018   "cima 4647   1stc1st 6162   2ndc2nd 6163   Q.cnq 7308   <Q cltq 7313   P.cnp 7319   <P cltp 7323

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-2o 6441  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-enq0 7452  df-nq0 7453  df-0nq0 7454  df-plq0 7455  df-mq0 7456  df-inp 7494  df-iltp 7498

This theorem is referenced by:  suplocsrlempr  7835