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Theorem sup00 6465
Description: The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
sup00  |-  sup ( B ,  (/) ,  R
)  =  (/)

Proof of Theorem sup00
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 6446 . 2  |-  sup ( B ,  (/) ,  R
)  =  U. {
x  e.  (/)  |  ( A. y  e.  B  -.  x R y  /\  A. y  e.  (/)  ( y R x  ->  E. z  e.  B  y R
z ) ) }
2 rab0 3274 . . 3  |-  { x  e.  (/)  |  ( A. y  e.  B  -.  x R y  /\  A. y  e.  (/)  ( y R x  ->  E. z  e.  B  y R
z ) ) }  =  (/)
32unieqi 3613 . 2  |-  U. {
x  e.  (/)  |  ( A. y  e.  B  -.  x R y  /\  A. y  e.  (/)  ( y R x  ->  E. z  e.  B  y R
z ) ) }  =  U. (/)
4 uni0 3630 . 2  |-  U. (/)  =  (/)
51, 3, 43eqtri 2106 1  |-  sup ( B ,  (/) ,  R
)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285   A.wral 2349   E.wrex 2350   {crab 2353   (/)c0 3252   U.cuni 3603   class class class wbr 3787   supcsup 6444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-nul 3253  df-sn 3406  df-uni 3604  df-sup 6446
This theorem is referenced by:  inf00  6493
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