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

Metamath Proof Explorer

Theorem sotr3

Description: Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021)

Ref Expression
Assertion sotr3 
|- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( X R Y /\ -. Z R Y ) -> X R Z ) )

Proof

Step Hyp Ref Expression
1 simp3 
 |-  ( ( X e. A /\ Y e. A /\ Z e. A ) -> Z e. A )
2 simp2 
 |-  ( ( X e. A /\ Y e. A /\ Z e. A ) -> Y e. A )
3 1 2 jca 
 |-  ( ( X e. A /\ Y e. A /\ Z e. A ) -> ( Z e. A /\ Y e. A ) )
4 sotric 
 |-  ( ( R Or A /\ ( Z e. A /\ Y e. A ) ) -> ( Z R Y <-> -. ( Z = Y \/ Y R Z ) ) )
5 3 4 sylan2 
 |-  ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( Z R Y <-> -. ( Z = Y \/ Y R Z ) ) )
6 5 con2bid 
 |-  ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( Z = Y \/ Y R Z ) <-> -. Z R Y ) )
7 6 adantr 
 |-  ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( ( Z = Y \/ Y R Z ) <-> -. Z R Y ) )
8 breq2 
 |-  ( Z = Y -> ( X R Z <-> X R Y ) )
9 8 biimprcd 
 |-  ( X R Y -> ( Z = Y -> X R Z ) )
10 9 adantl 
 |-  ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( Z = Y -> X R Z ) )
11 sotr 
 |-  ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( X R Y /\ Y R Z ) -> X R Z ) )
12 11 expdimp 
 |-  ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( Y R Z -> X R Z ) )
13 10 12 jaod 
 |-  ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( ( Z = Y \/ Y R Z ) -> X R Z ) )
14 7 13 sylbird 
 |-  ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( -. Z R Y -> X R Z ) )
15 14 expimpd 
 |-  ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( X R Y /\ -. Z R Y ) -> X R Z ) )