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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
supgtoreq.2
supgtoreq.3
supgtoreq.4
supgtoreq.5
Assertion
Ref Expression
supgtoreq

Proof of Theorem supgtoreq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supgtoreq.4 . . . . 5
2 supgtoreq.1 . . . . . 6
3 supgtoreq.2 . . . . . . 7
4 supgtoreq.3 . . . . . . . 8
5 ne0i 3767 . . . . . . . . 9
61, 5syl 17 . . . . . . . 8
7 fisup2g 7993 . . . . . . . 8
82, 4, 6, 3, 7syl13anc 1266 . . . . . . 7
9 ssrexv 3526 . . . . . . 7
103, 8, 9sylc 62 . . . . . 6
112, 10supub 7982 . . . . 5
121, 11mpd 15 . . . 4
13 supgtoreq.5 . . . . 5
1413breq1d 4433 . . . 4
1512, 14mtbird 302 . . 3
16 fisupcl 7994 . . . . . . . 8
172, 4, 6, 3, 16syl13anc 1266 . . . . . . 7
183, 17sseldd 3465 . . . . . 6
1913, 18eqeltrd 2507 . . . . 5
203, 1sseldd 3465 . . . . 5
21 sotric 4800 . . . . 5
222, 19, 20, 21syl12anc 1262 . . . 4
23 orcom 388 . . . . . 6
24 eqcom 2431 . . . . . . 7
2524orbi2i 521 . . . . . 6
2623, 25bitri 252 . . . . 5
2726notbii 297 . . . 4
2822, 27syl6rbb 265 . . 3
2915, 28mtbird 302 . 2
3029notnotrd 116 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wo 369   wa 370   wceq 1437   wcel 1872   wne 2614  wral 2771  wrex 2772   wss 3436  c0 3761   class class class wbr 4423   wor 4773  cfn 7580  csup 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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