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Metamath Proof Explorer

Theorem alephfp2

Description: The aleph function has at least one fixed point. Proposition 11.18 of TakeutiZaring p. 104. See alephfp for an actual example of a fixed point. Compare the inequality alephle that holds in general. Note that if x is a fixed point, then alephalephaleph` ... aleph `x = x . (Contributed by NM, 6-Nov-2004) (Revised by Mario Carneiro, 15-May-2015)

Ref Expression
Assertion alephfp2 ∃ 𝑥 ∈ On ( ℵ ‘ 𝑥 ) = 𝑥

Proof

Step Hyp Ref Expression
1 alephsson ⊢ ran ℵ ⊆ On
2 eqid ⊢ ( rec ( ℵ , ω ) ↾ ω ) = ( rec ( ℵ , ω ) ↾ ω )
3 2 alephfplem4 ⊢ ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ∈ ran ℵ
4 1 3 sselii ⊢ ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ∈ On
5 2 alephfp ⊢ ( ℵ ‘ ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ) = ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω )
6 fveq2 ⊢ ( 𝑥 = ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) → ( ℵ ‘ 𝑥 ) = ( ℵ ‘ ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ) )
7 id ⊢ ( 𝑥 = ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) → 𝑥 = ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) )
8 6 7 eqeq12d ⊢ ( 𝑥 = ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) → ( ( ℵ ‘ 𝑥 ) = 𝑥 ↔ ( ℵ ‘ ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ) = ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ) )
9 8 rspcev ⊢ ( ( ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ∈ On ∧ ( ℵ ‘ ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ) = ∪ ( ( rec ( ℵ , ω ) ↾ ω ) “ ω ) ) → ∃ 𝑥 ∈ On ( ℵ ‘ 𝑥 ) = 𝑥 )
10 4 5 9 mp2an ⊢ ∃ 𝑥 ∈ On ( ℵ ‘ 𝑥 ) = 𝑥