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Theorem sup00 6677
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 6658 . 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 3309 . . 3  |-  { x  e.  (/)  |  ( A. y  e.  B  -.  x R y  /\  A. y  e.  (/)  ( y R x  ->  E. z  e.  B  y R
z ) ) }  =  (/)
32unieqi 3658 . 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 3675 . 2  |-  U. (/)  =  (/)
51, 3, 43eqtri 2112 1  |-  sup ( B ,  (/) ,  R
)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289   A.wral 2359   E.wrex 2360   {crab 2363   (/)c0 3284   U.cuni 3648   class class class wbr 3837   supcsup 6656
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-uni 3649  df-sup 6658
This theorem is referenced by:  inf00  6705
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