Theorem suprlubex 9131

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 8258	. . . 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 8256	. . . 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 7199	. 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 2202   A.wral 2510   E.wrex 2511   C_ wss 3200   class class class wbr 4088   Or wor 4392   supcsup 7180   RRcr 8030   < clt 8213

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-po 4393  df-iso 4394  df-xp 4731  df-iota 5286  df-riota 5970  df-sup 7182  df-pnf 8215  df-mnf 8216  df-ltxr 8218

This theorem is referenced by:  suprnubex  9132  suprzclex  9577