Theorem frege83d

Description: If the image of the union of U and V is a subset of the union of U and V , A is an element of U and B follows A in the transitive closure of R , then B is an element of the union of U and V . Similar to Proposition 83 of Frege1879 p. 65. Compare with frege83 . (Contributed by RP, 15-Jul-2020)

	Ref	Expression
Hypotheses	frege83d.r	$$⊢\phi \to R\in \mathrm{V}$$
	frege83d.a	$$⊢\phi \to A\in U$$
	frege83d.b	$$⊢\phi \to B\in \mathrm{V}$$
	frege83d.ab	$$⊢\phi \to A\mathrm{t+}\left(R\right)B$$
	frege83d.he	$$⊢\phi \to R\left[U\cup V\right]\subseteq U\cup V$$
Assertion	frege83d	$$⊢\phi \to B\in \left(U\cup V\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege83d.r	$$⊢\phi \to R\in \mathrm{V}$$
2		frege83d.a	$$⊢\phi \to A\in U$$
3		frege83d.b	$$⊢\phi \to B\in \mathrm{V}$$
4		frege83d.ab	$$⊢\phi \to A\mathrm{t+}\left(R\right)B$$
5		frege83d.he	$$⊢\phi \to R\left[U\cup V\right]\subseteq U\cup V$$
6		ssun1	$$⊢U\subseteq U\cup V$$
7	6 2	sseldi	$$⊢\phi \to A\in \left(U\cup V\right)$$
8	1 7 3 4 5	frege81d	$$⊢\phi \to B\in \left(U\cup V\right)$$