infsdomnn

Description: An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004) (Revised by Mario Carneiro, 27-Apr-2015) Avoid ax-pow . (Revised by BTernaryTau, 7-Jan-2025)

		Ref	Expression
	Assertion	infsdomnn	$⊢ (ω ≼ A \land B \in ω) \to B ≺ A$

Proof

Step	Hyp	Ref	Expression
1		nnfi	$⊢ B \in ω \to B \in Fin$
2	1	adantl	$⊢ (ω ≼ A \land B \in ω) \to B \in Fin$
3		reldom	$⊢ Rel ≼$
4	3	brrelex1i	$⊢ ω ≼ A \to ω \in V$
5		nnsdomg	$⊢ (ω \in V \land B \in ω) \to B ≺ ω$
6	4 5	sylan	$⊢ (ω ≼ A \land B \in ω) \to B ≺ ω$
7		simpl	$⊢ (ω ≼ A \land B \in ω) \to ω ≼ A$
8		sdomdomtrfi	$⊢ (B \in Fin \land B ≺ ω \land ω ≼ A) \to B ≺ A$
9	2 6 7 8	syl3anc	$⊢ (ω ≼ A \land B \in ω) \to B ≺ A$

Metamath Proof Explorer

Theorem infsdomnn

Proof