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Theorem supgtoreq 7995
Description: The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.)
Hypotheses
Ref Expression
supgtoreq.1  |-  ( ph  ->  R  Or  A )
supgtoreq.2  |-  ( ph  ->  B  C_  A )
supgtoreq.3  |-  ( ph  ->  B  e.  Fin )
supgtoreq.4  |-  ( ph  ->  C  e.  B )
supgtoreq.5  |-  ( ph  ->  S  =  sup ( B ,  A ,  R ) )
Assertion
Ref Expression
supgtoreq  |-  ( ph  ->  ( C R S  \/  C  =  S ) )

Proof of Theorem supgtoreq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supgtoreq.4 . . . . 5  |-  ( ph  ->  C  e.  B )
2 supgtoreq.1 . . . . . 6  |-  ( ph  ->  R  Or  A )
3 supgtoreq.2 . . . . . . 7  |-  ( ph  ->  B  C_  A )
4 supgtoreq.3 . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
5 ne0i 3767 . . . . . . . . 9  |-  ( C  e.  B  ->  B  =/=  (/) )
61, 5syl 17 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
7 fisup2g 7993 . . . . . . . 8  |-  ( ( R  Or  A  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  A ) )  ->  E. x  e.  B  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
82, 4, 6, 3, 7syl13anc 1266 . . . . . . 7  |-  ( ph  ->  E. x  e.  B  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
9 ssrexv 3526 . . . . . . 7  |-  ( B 
C_  A  ->  ( E. x  e.  B  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) )  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) ) )
103, 8, 9sylc 62 . . . . . 6  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
112, 10supub 7982 . . . . 5  |-  ( ph  ->  ( C  e.  B  ->  -.  sup ( B ,  A ,  R
) R C ) )
121, 11mpd 15 . . . 4  |-  ( ph  ->  -.  sup ( B ,  A ,  R
) R C )
13 supgtoreq.5 . . . . 5  |-  ( ph  ->  S  =  sup ( B ,  A ,  R ) )
1413breq1d 4433 . . . 4  |-  ( ph  ->  ( S R C  <->  sup ( B ,  A ,  R ) R C ) )
1512, 14mtbird 302 . . 3  |-  ( ph  ->  -.  S R C )
16 fisupcl 7994 . . . . . . . 8  |-  ( ( R  Or  A  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  A ) )  ->  sup ( B ,  A ,  R )  e.  B
)
172, 4, 6, 3, 16syl13anc 1266 . . . . . . 7  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  B )
183, 17sseldd 3465 . . . . . 6  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
1913, 18eqeltrd 2507 . . . . 5  |-  ( ph  ->  S  e.  A )
203, 1sseldd 3465 . . . . 5  |-  ( ph  ->  C  e.  A )
21 sotric 4800 . . . . 5  |-  ( ( R  Or  A  /\  ( S  e.  A  /\  C  e.  A
) )  ->  ( S R C  <->  -.  ( S  =  C  \/  C R S ) ) )
222, 19, 20, 21syl12anc 1262 . . . 4  |-  ( ph  ->  ( S R C  <->  -.  ( S  =  C  \/  C R S ) ) )
23 orcom 388 . . . . . 6  |-  ( ( S  =  C  \/  C R S )  <->  ( C R S  \/  S  =  C ) )
24 eqcom 2431 . . . . . . 7  |-  ( S  =  C  <->  C  =  S )
2524orbi2i 521 . . . . . 6  |-  ( ( C R S  \/  S  =  C )  <->  ( C R S  \/  C  =  S )
)
2623, 25bitri 252 . . . . 5  |-  ( ( S  =  C  \/  C R S )  <->  ( C R S  \/  C  =  S ) )
2726notbii 297 . . . 4  |-  ( -.  ( S  =  C  \/  C R S )  <->  -.  ( C R S  \/  C  =  S ) )
2822, 27syl6rbb 265 . . 3  |-  ( ph  ->  ( -.  ( C R S  \/  C  =  S )  <->  S R C ) )
2915, 28mtbird 302 . 2  |-  ( ph  ->  -.  -.  ( C R S  \/  C  =  S ) )
3029notnotrd 116 1  |-  ( ph  ->  ( C R S  \/  C  =  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772    C_ wss 3436   (/)c0 3761   class class class wbr 4423    Or wor 4773   Fincfn 7580   supcsup 7963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-om 6707  df-1o 7193  df-er 7374  df-en 7581  df-fin 7584  df-sup 7965
This theorem is referenced by:  infltoreq  8027  supfirege  10605
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