Theorem sup00 7170

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.

Step	Hyp	Ref	Expression
1		df-sup 7151	. 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 3520	. . 3 |- { x e. (/) | ( A. y e. B -. x R y /\ A. y e. (/) ( y R x -> E. z e. B y R z ) ) } = (/)
3	2	unieqi 3898	. 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 3915	. 2 |- U. (/) = (/)
5	1, 3, 4	3eqtri 2254	1 |- sup ( B , (/) , R ) = (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   A.wral 2508   E.wrex 2509   {crab 2512   (/)c0 3491   U.cuni 3888   class class class wbr 4083   supcsup 7149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-uni 3889  df-sup 7151

This theorem is referenced by:  inf00  7198