This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem falseral0OLD

Description: Obsolete version of falseral0 as of 16-Feb-2026. (Contributed by AV, 30-Oct-2020) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion falseral0OLD 
|- ( ( A. x -. ph /\ A. x e. A ph ) -> A = (/) )

Proof

Step Hyp Ref Expression
1 df-ral 
 |-  ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
2 19.26 
 |-  ( A. x ( -. ph /\ ( x e. A -> ph ) ) <-> ( A. x -. ph /\ A. x ( x e. A -> ph ) ) )
3 con3 
 |-  ( ( x e. A -> ph ) -> ( -. ph -> -. x e. A ) )
4 3 impcom 
 |-  ( ( -. ph /\ ( x e. A -> ph ) ) -> -. x e. A )
5 4 alimi 
 |-  ( A. x ( -. ph /\ ( x e. A -> ph ) ) -> A. x -. x e. A )
6 alnex 
 |-  ( A. x -. x e. A <-> -. E. x x e. A )
7 5 6 sylib 
 |-  ( A. x ( -. ph /\ ( x e. A -> ph ) ) -> -. E. x x e. A )
8 notnotb 
 |-  ( A = (/) <-> -. -. A = (/) )
9 neq0 
 |-  ( -. A = (/) <-> E. x x e. A )
10 8 9 xchbinx 
 |-  ( A = (/) <-> -. E. x x e. A )
11 7 10 sylibr 
 |-  ( A. x ( -. ph /\ ( x e. A -> ph ) ) -> A = (/) )
12 2 11 sylbir 
 |-  ( ( A. x -. ph /\ A. x ( x e. A -> ph ) ) -> A = (/) )
13 1 12 sylan2b 
 |-  ( ( A. x -. ph /\ A. x e. A ph ) -> A = (/) )