Metamath Proof Explorer

Theorem sdomne0d

Description: A class that strictly dominates any set is not empty. (Contributed by RP, 3-Sep-2024)

Ref Expression
Hypotheses sdomne0d.a 
|- ( ph -> B ~< A )
sdomne0d.b 
|- ( ph -> B e. V )
Assertion sdomne0d 
|- ( ph -> A =/= (/) )

Proof

Step Hyp Ref Expression
1 sdomne0d.a 
 |-  ( ph -> B ~< A )
2 sdomne0d.b 
 |-  ( ph -> B e. V )
3 breq1 
 |-  ( B = (/) -> ( B ~< A <-> (/) ~< A ) )
4 3 biimpd 
 |-  ( B = (/) -> ( B ~< A -> (/) ~< A ) )
5 4 a1i 
 |-  ( B e. V -> ( B = (/) -> ( B ~< A -> (/) ~< A ) ) )
6 0sdomg 
 |-  ( B e. V -> ( (/) ~< B <-> B =/= (/) ) )
7 sdomtr 
 |-  ( ( (/) ~< B /\ B ~< A ) -> (/) ~< A )
8 7 ex 
 |-  ( (/) ~< B -> ( B ~< A -> (/) ~< A ) )
9 6 8 biimtrrdi 
 |-  ( B e. V -> ( B =/= (/) -> ( B ~< A -> (/) ~< A ) ) )
10 5 9 pm2.61dne 
 |-  ( B e. V -> ( B ~< A -> (/) ~< A ) )
11 2 10 syl 
 |-  ( ph -> ( B ~< A -> (/) ~< A ) )
12 relsdom 
 |-  Rel ~<
13 12 brrelex2i 
 |-  ( (/) ~< A -> A e. _V )
14 0sdomg 
 |-  ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
15 13 14 syl 
 |-  ( (/) ~< A -> ( (/) ~< A <-> A =/= (/) ) )
16 15 ibi 
 |-  ( (/) ~< A -> A =/= (/) )
17 11 16 syl6 
 |-  ( ph -> ( B ~< A -> A =/= (/) ) )
18 1 17 mpd 
 |-  ( ph -> A =/= (/) )