Theorem alephgeom

Description: Every aleph is greater than or equal to the set of natural numbers. (Contributed by NM, 11-Nov-2003)

	Ref	Expression
Assertion	alephgeom	⊢ ( 𝐴 ∈ On ↔ ω ⊆ ( ℵ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		aleph0	⊢ ( ℵ ‘ ∅ ) = ω
2		0ss	⊢ ∅ ⊆ 𝐴
3		0elon	⊢ ∅ ∈ On
4		alephord3	⊢ ( ( ∅ ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ⊆ 𝐴 ↔ ( ℵ ‘ ∅ ) ⊆ ( ℵ ‘ 𝐴 ) ) )
5	3 4	mpan	⊢ ( 𝐴 ∈ On → ( ∅ ⊆ 𝐴 ↔ ( ℵ ‘ ∅ ) ⊆ ( ℵ ‘ 𝐴 ) ) )
6	2 5	mpbii	⊢ ( 𝐴 ∈ On → ( ℵ ‘ ∅ ) ⊆ ( ℵ ‘ 𝐴 ) )
7	1 6	eqsstrrid	⊢ ( 𝐴 ∈ On → ω ⊆ ( ℵ ‘ 𝐴 ) )
8		peano1	⊢ ∅ ∈ ω
9		ordom	⊢ Ord ω
10		ord0	⊢ Ord ∅
11		ordtri1	⊢ ( ( Ord ω ∧ Ord ∅ ) → ( ω ⊆ ∅ ↔ ¬ ∅ ∈ ω ) )
12	9 10 11	mp2an	⊢ ( ω ⊆ ∅ ↔ ¬ ∅ ∈ ω )
13	12	con2bii	⊢ ( ∅ ∈ ω ↔ ¬ ω ⊆ ∅ )
14	8 13	mpbi	⊢ ¬ ω ⊆ ∅
15		ndmfv	⊢ ( ¬ 𝐴 ∈ dom ℵ → ( ℵ ‘ 𝐴 ) = ∅ )
16	15	sseq2d	⊢ ( ¬ 𝐴 ∈ dom ℵ → ( ω ⊆ ( ℵ ‘ 𝐴 ) ↔ ω ⊆ ∅ ) )
17	14 16	mtbiri	⊢ ( ¬ 𝐴 ∈ dom ℵ → ¬ ω ⊆ ( ℵ ‘ 𝐴 ) )
18	17	con4i	⊢ ( ω ⊆ ( ℵ ‘ 𝐴 ) → 𝐴 ∈ dom ℵ )
19		alephfnon	⊢ ℵ Fn On
20		fndm	⊢ ( ℵ Fn On → dom ℵ = On )
21	19 20	ax-mp	⊢ dom ℵ = On
22	18 21	eleqtrdi	⊢ ( ω ⊆ ( ℵ ‘ 𝐴 ) → 𝐴 ∈ On )
23	7 22	impbii	⊢ ( 𝐴 ∈ On ↔ ω ⊆ ( ℵ ‘ 𝐴 ) )