Theorem domtrfir

Description: Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr ). (Contributed by BTernaryTau, 24-Nov-2024)

	Ref	Expression
Assertion	domtrfir	|- ( ( C e. Fin /\ A ~<_ B /\ B ~<_ C ) -> A ~<_ C )

Proof

Step	Hyp	Ref	Expression
1		domfi	|- ( ( C e. Fin /\ B ~<_ C ) -> B e. Fin )
2	1	3adant2	|- ( ( C e. Fin /\ A ~<_ B /\ B ~<_ C ) -> B e. Fin )
3		domtrfi	|- ( ( B e. Fin /\ A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
4	2 3	syld3an1	|- ( ( C e. Fin /\ A ~<_ B /\ B ~<_ C ) -> A ~<_ C )