Theorem suplocexpr 7840

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 4048	. . . . . 6 |- ( a = w -> ( a <Q u <-> w <Q u ) )
5	4	cbvrexv 2739	. . . . 5 |- ( E. a e. |^| ( 2nd " A ) a <Q u <-> E. w e. |^| ( 2nd " A ) w <Q u )
6	5	rabbii 2758	. . . 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 3823	. . 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 7837	. 2 |- ( ph -> <. U. ( 1st " A ) , { u e. Q. | E. a e. |^| ( 2nd " A ) a <Q u } >. e. P. )
9	1, 2, 3, 7	suplocexprlemub 7838	. 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 7839	. . 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 2580	. 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 4048	. . . . . 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 669	. . . . 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 2506	. . . 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 4049	. . . . . 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 2506	. . . 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 2877	. 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 1248	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 710   = wceq 1373   E.wex 1515   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   <.cop 3636   U.cuni 3850   |^|cint 3885   class class class wbr 4045   "cima 4679   1stc1st 6226   2ndc2nd 6227   Q.cnq 7395   <Q cltq 7400   P.cnp 7406   <P cltp 7410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-eprel 4337  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-1o 6504  df-2o 6505  df-oadd 6508  df-omul 6509  df-er 6622  df-ec 6624  df-qs 6628  df-ni 7419  df-pli 7420  df-mi 7421  df-lti 7422  df-plpq 7459  df-mpq 7460  df-enq 7462  df-nqqs 7463  df-plqqs 7464  df-mqqs 7465  df-1nqqs 7466  df-rq 7467  df-ltnqqs 7468  df-enq0 7539  df-nq0 7540  df-0nq0 7541  df-plq0 7542  df-mq0 7543  df-inp 7581  df-iltp 7585

This theorem is referenced by:  suplocsrlempr  7922