Theorem trwf

Description: The class of well-founded sets is transitive. (Contributed by Eric Schmidt, 9-Sep-2025)

	Ref	Expression
Assertion	trwf	⊢ Tr ∪ ( 𝑅1 “ On )

Proof

Step	Hyp	Ref	Expression
1		r1elssi	⊢ ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) → 𝑥 ⊆ ∪ ( 𝑅1 “ On ) )
2	1	rgen	⊢ ∀ 𝑥 ∈ ∪ ( 𝑅1 “ On ) 𝑥 ⊆ ∪ ( 𝑅1 “ On )
3		dftr3	⊢ ( Tr ∪ ( 𝑅1 “ On ) ↔ ∀ 𝑥 ∈ ∪ ( 𝑅1 “ On ) 𝑥 ⊆ ∪ ( 𝑅1 “ On ) )
4	2 3	mpbir	⊢ Tr ∪ ( 𝑅1 “ On )