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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

Proof of Theorem sup00
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 6446 . 2
2 rab0 3274 . . 3
32unieqi 3613 . 2
4 uni0 3630 . 2
51, 3, 43eqtri 2106 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102   wceq 1285  wral 2349  wrex 2350  crab 2353  c0 3252  cuni 3603   class class class wbr 3787  csup 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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