Theorem suprubex 8995

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 3185	. 2 |- ( ph -> B e. RR )
4		lttri3 8123	. . . 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 7073	. 2 |- ( ph -> sup ( A , RR , < ) e. RR )
8	5, 6	supubti 7074	. . 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 8164	1 |- ( ph -> B <_ sup ( A , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   class class class wbr 4034   supcsup 7057   RRcr 7895   < clt 8078   <_ cle 8079

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-pre-ltirr 8008  ax-pre-apti 8011

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-iota 5220  df-riota 5880  df-sup 7059  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084

This theorem is referenced by:  suprzclex  9441