Theorem suprlubex 8973

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 8101	. . . 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 8099	. . . 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 7062	. 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 2164   A.wral 2472   E.wrex 2473   C_ wss 3154   class class class wbr 4030   Or wor 4327   supcsup 7043   RRcr 7873   < clt 8056

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-po 4328  df-iso 4329  df-xp 4666  df-iota 5216  df-riota 5874  df-sup 7045  df-pnf 8058  df-mnf 8059  df-ltxr 8061

This theorem is referenced by:  suprnubex  8974  suprzclex  9418