Theorem suprlubex 9120

Description: The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)

Hypotheses

Ref	Expression
suprubex.ex	|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
suprubex.ss	|- ( ph -> A C_ RR )
suprlubex.b	|- ( ph -> B e. RR )

Assertion

Ref	Expression
suprlubex	|- ( ph -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )

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

Allowed substitution hints:   ph( y, z)   B( x, y)

Proof of Theorem suprlubex

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suprlubex.b	. 2 |- ( ph -> B e. RR )
2		lttri3 8247	. . . 4 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3	2	adantl 277	. . 3 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
4		suprubex.ex	. . 3 |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
5		ltso 8245	. . . 4 |- < Or RR
6	5	a1i 9	. . 3 |- ( ph -> < Or RR )
7		suprubex.ss	. . 3 |- ( ph -> A C_ RR )
8	3, 4, 6, 7	suplub2ti 7189	. 2 |- ( ( ph /\ B e. RR ) -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )
9	1, 8	mpdan 421	1 |- ( ph -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   A.wral 2508   E.wrex 2509   C_ wss 3198   class class class wbr 4084   Or wor 4388   supcsup 7170   RRcr 8019   < clt 8202

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-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135

This theorem depends on definitions:  df-bi 117  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-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-br 4085  df-opab 4147  df-po 4389  df-iso 4390  df-xp 4727  df-iota 5282  df-riota 5964  df-sup 7172  df-pnf 8204  df-mnf 8205  df-ltxr 8207

This theorem is referenced by:  suprnubex  9121  suprzclex  9566