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Theorem sup0 7980
 Description: The supremum of an empty set under a base set which has a unique smallest element is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
sup0
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem sup0
StepHypRef Expression
1 sup0riota 7979 . . 3
213ad2ant1 1029 . 2
3 simp2r 1035 . . 3
4 simpl 459 . . . . . 6
54anim1i 572 . . . . 5
653adant1 1026 . . . 4
7 breq2 4406 . . . . . . 7
87notbid 296 . . . . . 6
98ralbidv 2827 . . . . 5
109riota2 6274 . . . 4
116, 10syl 17 . . 3
123, 11mpbid 214 . 2
132, 12eqtrd 2485 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 188   wa 371   w3a 985   wceq 1444   wcel 1887  wral 2737  wreu 2739  c0 3731   class class class wbr 4402   wor 4754  crio 6251  csup 7954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-po 4755  df-so 4756  df-iota 5546  df-riota 6252  df-sup 7956 This theorem is referenced by:  infempty  8022
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