Theorem frege100

Description: One direction of dffrege99 . Proposition 100 of Frege1879 p. 72. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege99.z	$$⊢Z\in U$$
Assertion	frege100	$$⊢X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege99.z	$$⊢Z\in U$$
2	1	dffrege99	$$⊢\left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\leftrightarrow X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z$$
3		frege57aid	$$⊢\left(\left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\leftrightarrow X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)\to \left(X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\right)$$
4	2 3	ax-mp	$$⊢X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)$$