ispod

Description: Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014)

	Ref	Expression
Hypotheses	ispod.1	$⊢ (φ \land x \in A) \to \neg x R x$
	ispod.2	$⊢ (φ \land (x \in A \land y \in A \land z \in A)) \to ((x R y \land y R z) \to x R z)$
Assertion	ispod	$⊢ φ \to R Po A$

Step	Hyp	Ref	Expression
1		ispod.1	$⊢ (φ \land x \in A) \to \neg x R x$
2		ispod.2	$⊢ (φ \land (x \in A \land y \in A \land z \in A)) \to ((x R y \land y R z) \to x R z)$
3	1	3ad2antr1	$⊢ (φ \land (x \in A \land y \in A \land z \in A)) \to \neg x R x$
4	3 2	jca	$⊢ (φ \land (x \in A \land y \in A \land z \in A)) \to (\neg x R x \land ((x R y \land y R z) \to x R z))$
5	4	ralrimivvva	$⊢ φ \to \forall x \in A \forall y \in A \forall z \in A (\neg x R x \land ((x R y \land y R z) \to x R z))$
6		df-po	$⊢ R Po A \leftrightarrow \forall x \in A \forall y \in A \forall z \in A (\neg x R x \land ((x R y \land y R z) \to x R z))$
7	5 6	sylibr	$⊢ φ \to R Po A$

Metamath Proof Explorer