Theorem sup00 7201

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 7182	. 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 3523	. . 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 3903	. 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 3920	. 2 |- U. (/) = (/)
5	1, 3, 4	3eqtri 2256	1 |- sup ( B , (/) , R ) = (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1397   A.wral 2510   E.wrex 2511   {crab 2514   (/)c0 3494   U.cuni 3893   class class class wbr 4088   supcsup 7180

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-uni 3894  df-sup 7182

This theorem is referenced by:  inf00  7229