Theorem sup00 8370

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 8348	. 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 3955	. . 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 4445	. 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 4465	. 2 |- U. (/) = (/)
5	1, 3, 4	3eqtri 2648	1 |- sup ( B , (/) , R ) = (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   E.wrex 2913   {crab 2916   (/)c0 3915   U.cuni 4436   class class class wbr 4653   supcsup 8346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437  df-sup 8348

This theorem is referenced by:  inf00  8411