Theorem suprubex 9024

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 3194	. 2 |- ( ph -> B e. RR )
4		lttri3 8152	. . . 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 7100	. 2 |- ( ph -> sup ( A , RR , < ) e. RR )
8	5, 6	supubti 7101	. . 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 8193	1 |- ( ph -> B <_ sup ( A , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   A.wral 2484   E.wrex 2485   C_ wss 3166   class class class wbr 4044   supcsup 7084   RRcr 7924   < clt 8107   <_ cle 8108

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037  ax-pre-apti 8040

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-iota 5232  df-riota 5899  df-sup 7086  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113

This theorem is referenced by:  suprzclex  9471