Theorem mulsne0bd

Description: The product of two non-zero surreals is non-zero. (Contributed by Scott Fenton, 16-Apr-2025)

	Ref	Expression
Hypotheses	muls0ord.1	|- ( ph -> A e. No )
	muls0ord.2	|- ( ph -> B e. No )
Assertion	mulsne0bd	|- ( ph -> ( ( A x.s B ) =/= 0s <-> ( A =/= 0s /\ B =/= 0s ) ) )

Proof

Step	Hyp	Ref	Expression
1		muls0ord.1	|- ( ph -> A e. No )
2		muls0ord.2	|- ( ph -> B e. No )
3	1 2	muls0ord	|- ( ph -> ( ( A x.s B ) = 0s <-> ( A = 0s \/ B = 0s ) ) )
4	3	necon3abid	|- ( ph -> ( ( A x.s B ) =/= 0s <-> -. ( A = 0s \/ B = 0s ) ) )
5		neanior	|- ( ( A =/= 0s /\ B =/= 0s ) <-> -. ( A = 0s \/ B = 0s ) )
6	4 5	bitr4di	|- ( ph -> ( ( A x.s B ) =/= 0s <-> ( A =/= 0s /\ B =/= 0s ) ) )