Theorem antisymrelressn

Description: (Contributed by Peter Mazsa, 29-Jun-2024)

	Ref	Expression
Assertion	antisymrelressn	$$⊢AntisymRel\left({R\upharpoonright }_{\left\{A\right\}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		antisymressn	$$⊢\forall x\forall y\left(\left(x\left({R\upharpoonright }_{\left\{A\right\}}\right)y\wedge y\left({R\upharpoonright }_{\left\{A\right\}}\right)x\right)\to x=y\right)$$
2		relres	$$⊢\mathrm{Rel}\left({R\upharpoonright }_{\left\{A\right\}}\right)$$
3		dfantisymrel5	$$⊢AntisymRel\left({R\upharpoonright }_{\left\{A\right\}}\right)\leftrightarrow \left(\forall x\forall y\left(\left(x\left({R\upharpoonright }_{\left\{A\right\}}\right)y\wedge y\left({R\upharpoonright }_{\left\{A\right\}}\right)x\right)\to x=y\right)\wedge \mathrm{Rel}\left({R\upharpoonright }_{\left\{A\right\}}\right)\right)$$
4	1 2 3	mpbir2an	$$⊢AntisymRel\left({R\upharpoonright }_{\left\{A\right\}}\right)$$