Theorem dalemccnedd

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012)

	Ref	Expression
Hypothesis	da.ps0	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
Assertion	dalemccnedd	$$⊢\psi \to c\ne d$$

Proof

Step	Hyp	Ref	Expression
1		da.ps0	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
2		simp31	$$⊢\left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)\to d\ne c$$
3	1 2	sylbi	$$⊢\psi \to d\ne c$$
4	3	necomd	$$⊢\psi \to c\ne d$$