Theorem suprubex 9281

Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-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 )
suprubex.b	|- ( ph -> B e. A )

Assertion

Ref	Expression
suprubex	|- ( ph -> B <_ sup ( A , RR , < ) )

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

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

Proof of Theorem suprubex

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

Step	Hyp	Ref	Expression
1		suprubex.ss	. . 3 |- ( ph -> A C_ RR )
2		suprubex.b	. . 3 |- ( ph -> B e. A )
3	1, 2	sseldd 3249	. 2 |- ( ph -> B e. RR )
4		lttri3 8405	. . . 4 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
5	4	adantl 277	. . 3 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
6		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 ) ) )
7	5, 6	supclti 7338	. 2 |- ( ph -> sup ( A , RR , < ) e. RR )
8	5, 6	supubti 7339	. . 3 |- ( ph -> ( B e. A -> -. sup ( A , RR , < ) < B ) )
9	2, 8	mpd 13	. 2 |- ( ph -> -. sup ( A , RR , < ) < B )
10	3, 7, 9	nltled 8447	1 |- ( ph -> B <_ sup ( A , RR , < ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   class class class wbr 4130   supcsup 7322   RRcr 8178   < clt 8360   <_ cle 8361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-apti 8294

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-iota 5337  df-riota 6038  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366

This theorem is used by:  suprzclex  9744