Theorem hdmaplem1

Description: Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015)

	Ref	Expression
Hypotheses	hdmaplem1.v	$$⊢V={\mathrm{Base}}_W$$
	hdmaplem1.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
	hdmaplem1.w	$$⊢\phi \to W\in \mathrm{LMod}$$
	hdmaplem1.z	$$⊢\phi \to Z\in V$$
	hdmaplem1.j	$$⊢\phi \to \neg Z\in \left(N\left(\left\{X\right\}\right)\cup N\left(\left\{Y\right\}\right)\right)$$
	hdmaplem1.x	$$⊢\phi \to X\in V$$
Assertion	hdmaplem1	$$⊢\phi \to N\left(\left\{Z\right\}\right)\ne N\left(\left\{X\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		hdmaplem1.v	$$⊢V={\mathrm{Base}}_W$$
2		hdmaplem1.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
3		hdmaplem1.w	$$⊢\phi \to W\in \mathrm{LMod}$$
4		hdmaplem1.z	$$⊢\phi \to Z\in V$$
5		hdmaplem1.j	$$⊢\phi \to \neg Z\in \left(N\left(\left\{X\right\}\right)\cup N\left(\left\{Y\right\}\right)\right)$$
6		hdmaplem1.x	$$⊢\phi \to X\in V$$
7		elun1	$$⊢Z\in N\left(\left\{X\right\}\right)\to Z\in \left(N\left(\left\{X\right\}\right)\cup N\left(\left\{Y\right\}\right)\right)$$
8	5 7	nsyl	$$⊢\phi \to \neg Z\in N\left(\left\{X\right\}\right)$$
9	1 2 3 4 6 8	lspsnne2	$$⊢\phi \to N\left(\left\{Z\right\}\right)\ne N\left(\left\{X\right\}\right)$$