Theorem alephprc

Description: The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of TakeutiZaring p. 90. (Contributed by NM, 11-Nov-2003)

	Ref	Expression
Assertion	alephprc	⊢ ¬ ran ℵ ∈ V

Proof

Step	Hyp	Ref	Expression
1		cardprc	⊢ { 𝑥 ∣ ( card ‘ 𝑥 ) = 𝑥 } ∉ V
2	1	neli	⊢ ¬ { 𝑥 ∣ ( card ‘ 𝑥 ) = 𝑥 } ∈ V
3		cardnum	⊢ { 𝑥 ∣ ( card ‘ 𝑥 ) = 𝑥 } = ( ω ∪ ran ℵ )
4	3	eleq1i	⊢ ( { 𝑥 ∣ ( card ‘ 𝑥 ) = 𝑥 } ∈ V ↔ ( ω ∪ ran ℵ ) ∈ V )
5	2 4	mtbi	⊢ ¬ ( ω ∪ ran ℵ ) ∈ V
6		omex	⊢ ω ∈ V
7		unexg	⊢ ( ( ω ∈ V ∧ ran ℵ ∈ V ) → ( ω ∪ ran ℵ ) ∈ V )
8	6 7	mpan	⊢ ( ran ℵ ∈ V → ( ω ∪ ran ℵ ) ∈ V )
9	5 8	mto	⊢ ¬ ran ℵ ∈ V