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Theorem sup0riota 7997
 Description: The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
sup0riota
Distinct variable groups:   ,,   ,,

Proof of Theorem sup0riota
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3
21supval2 7987 . 2
3 ral0 3865 . . . . . 6
43biantrur 514 . . . . 5
5 rex0 3737 . . . . . . 7
6 imnot 347 . . . . . . 7
75, 6ax-mp 5 . . . . . 6
87ralbii 2823 . . . . 5
94, 8bitr3i 259 . . . 4
109a1i 11 . . 3
1110riotabidv 6272 . 2
122, 11eqtrd 2505 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wa 376   wceq 1452  wral 2756  wrex 2757  c0 3722   class class class wbr 4395   wor 4759  crio 6269  csup 7972 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-po 4760  df-so 4761  df-iota 5553  df-riota 6270  df-sup 7974 This theorem is referenced by:  sup0  7998
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