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Metamath Proof Explorer

Theorem php5

Description: Corollary of the Pigeonhole Principle php : a natural number is not equinumerous to its successor. Corollary 10.21(1) of TakeutiZaring p. 90. (Contributed by NM, 26-Jul-2004)

		Ref	Expression
	Assertion	php5	⊢ ( 𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		php4	⊢ ( 𝐴 ∈ ω → 𝐴 ≺ suc 𝐴 )
2		sdomnen	⊢ ( 𝐴 ≺ suc 𝐴 → ¬ 𝐴 ≈ suc 𝐴 )
3	1 2	syl	⊢ ( 𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴 )