Theorem eldisjn0el

Description: Special case of disjdmqseq (perhaps this is the closest theorem to the former prter2 ). (Contributed by Peter Mazsa, 26-Sep-2021)

	Ref	Expression
Assertion	eldisjn0el	|- ( ElDisj A -> ( -. (/) e. A <-> ( U. A /. ~ A ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		disjdmqseq	|- ( Disj ( `' _E |` A ) -> ( ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A <-> ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) )
2		df-eldisj	|- ( ElDisj A <-> Disj ( `' _E |` A ) )
3		n0el3	|- ( -. (/) e. A <-> ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A )
4		dmqs1cosscnvepreseq	|- ( ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A <-> ( U. A /. ~ A ) = A )
5	4	bicomi	|- ( ( U. A /. ~ A ) = A <-> ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A )
6	3 5	bibi12i	|- ( ( -. (/) e. A <-> ( U. A /. ~ A ) = A ) <-> ( ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A <-> ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) )
7	1 2 6	3imtr4i	|- ( ElDisj A -> ( -. (/) e. A <-> ( U. A /. ~ A ) = A ) )