Theorem bnj98

Description: Technical lemma for bnj150 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	bnj98	$$⊢\forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in 1_{𝑜}\to F\left(\mathrm{suc}i\right)=\bigcup _{y\in F\left(i\right)}pred\left(y,A,R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢i\in \mathrm{V}$$
2	1	sucid	$$⊢i\in \mathrm{suc}i$$
3	2	n0ii	$$⊢\neg \mathrm{suc}i=\varnothing$$
4		df-suc	$$⊢\mathrm{suc}i=i\cup \left\{i\right\}$$
5		df-un	$$⊢i\cup \left\{i\right\}=\left\{x|\left(x\in i\vee x\in \left\{i\right\}\right)\right\}$$
6	4 5	eqtri	$$⊢\mathrm{suc}i=\left\{x|\left(x\in i\vee x\in \left\{i\right\}\right)\right\}$$
7		df1o2	$$⊢1_{𝑜}=\left\{\varnothing \right\}$$
8	6 7	eleq12i	$$⊢\mathrm{suc}i\in 1_{𝑜}\leftrightarrow \left\{x|\left(x\in i\vee x\in \left\{i\right\}\right)\right\}\in \left\{\varnothing \right\}$$
9		elsni	$$⊢\left\{x|\left(x\in i\vee x\in \left\{i\right\}\right)\right\}\in \left\{\varnothing \right\}\to \left\{x|\left(x\in i\vee x\in \left\{i\right\}\right)\right\}=\varnothing$$
10	8 9	sylbi	$$⊢\mathrm{suc}i\in 1_{𝑜}\to \left\{x|\left(x\in i\vee x\in \left\{i\right\}\right)\right\}=\varnothing$$
11	6 10	syl5eq	$$⊢\mathrm{suc}i\in 1_{𝑜}\to \mathrm{suc}i=\varnothing$$
12	3 11	mto	$$⊢\neg \mathrm{suc}i\in 1_{𝑜}$$
13	12	pm2.21i	$$⊢\mathrm{suc}i\in 1_{𝑜}\to F\left(\mathrm{suc}i\right)=\bigcup _{y\in F\left(i\right)}pred\left(y,A,R\right)$$
14	13	rgenw	$$⊢\forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in 1_{𝑜}\to F\left(\mathrm{suc}i\right)=\bigcup _{y\in F\left(i\right)}pred\left(y,A,R\right)\right)$$