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Theorem sup00 6939
 Description: The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
sup00

Proof of Theorem sup00
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 6920 . 2
2 rab0 3422 . . 3
32unieqi 3782 . 2
4 uni0 3799 . 2
51, 3, 43eqtri 2182 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wceq 1335  wral 2435  wrex 2436  crab 2439  c0 3394  cuni 3772   class class class wbr 3965  csup 6918 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-uni 3773  df-sup 6920 This theorem is referenced by:  inf00  6967
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