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Theorem sup00 6856
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 6837 . 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 3359 . . 3  |-  { x  e.  (/)  |  ( A. y  e.  B  -.  x R y  /\  A. y  e.  (/)  ( y R x  ->  E. z  e.  B  y R
z ) ) }  =  (/)
32unieqi 3714 . 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 3731 . 2  |-  U. (/)  =  (/)
51, 3, 43eqtri 2140 1  |-  sup ( B ,  (/) ,  R
)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1314   A.wral 2391   E.wrex 2392   {crab 2395   (/)c0 3331   U.cuni 3704   class class class wbr 3897   supcsup 6835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-uni 3705  df-sup 6837
This theorem is referenced by:  inf00  6884
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