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Metamath Proof Explorer

Theorem om0x

Description: Ordinal multiplication with zero. Definition 8.15 of TakeutiZaring p. 62. Unlike om0 , this version works whether or not A is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996) (New usage is discouraged.)

		Ref	Expression
	Assertion	om0x	⊢ ( 𝐴 ·_o ∅ ) = ∅

Proof

Step	Hyp	Ref	Expression
1		om0	⊢ ( 𝐴 ∈ On → ( 𝐴 ·_o ∅ ) = ∅ )
2	1	adantr	⊢ ( ( 𝐴 ∈ On ∧ ∅ ∈ On ) → ( 𝐴 ·_o ∅ ) = ∅ )
3		fnom	⊢ ·_o Fn ( On × On )
4	3	fndmi	⊢ dom ·_o = ( On × On )
5	4	ndmov	⊢ ( ¬ ( 𝐴 ∈ On ∧ ∅ ∈ On ) → ( 𝐴 ·_o ∅ ) = ∅ )
6	2 5	pm2.61i	⊢ ( 𝐴 ·_o ∅ ) = ∅