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Metamath Proof Explorer

Theorem ordn2lp

Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of BellMachover p. 469. (Contributed by NM, 3-Apr-1994)

		Ref	Expression
	Assertion	ordn2lp	⊢ ( Ord 𝐴 → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ordirr	⊢ ( Ord 𝐴 → ¬ 𝐴 ∈ 𝐴 )
2		ordtr	⊢ ( Ord 𝐴 → Tr 𝐴 )
3		trel	⊢ ( Tr 𝐴 → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) → 𝐴 ∈ 𝐴 ) )
4	2 3	syl	⊢ ( Ord 𝐴 → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) → 𝐴 ∈ 𝐴 ) )
5	1 4	mtod	⊢ ( Ord 𝐴 → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) )