Theorem imadisjlnd

Description: Natural deduction form of one negated side of imadisj . (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypothesis	imadisjlnd.1	$$⊢\phi \to \mathrm{dom}A\cap B\ne \varnothing$$
Assertion	imadisjlnd	$$⊢\phi \to A\left[B\right]\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		imadisjlnd.1	$$⊢\phi \to \mathrm{dom}A\cap B\ne \varnothing$$
2		imadisj	$$⊢A\left[B\right]=\varnothing \leftrightarrow \mathrm{dom}A\cap B=\varnothing$$
3	2	biimpi	$$⊢A\left[B\right]=\varnothing \to \mathrm{dom}A\cap B=\varnothing$$
4	3	necon3i	$$⊢\mathrm{dom}A\cap B\ne \varnothing \to A\left[B\right]\ne \varnothing$$
5	1 4	syl	$$⊢\phi \to A\left[B\right]\ne \varnothing$$