Theorem dju1en

Description: Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of Enderton p. 143. (Contributed by NM, 28-Sep-2004) (Revised by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	dju1en	|- ( ( A e. V /\ -. A e. A ) -> ( A |_| 1o ) ~~ suc A )

Proof

Step	Hyp	Ref	Expression
1		enrefg	|- ( A e. V -> A ~~ A )
2	1	adantr	|- ( ( A e. V /\ -. A e. A ) -> A ~~ A )
3		ensn1g	|- ( A e. V -> { A } ~~ 1o )
4	3	ensymd	|- ( A e. V -> 1o ~~ { A } )
5	4	adantr	|- ( ( A e. V /\ -. A e. A ) -> 1o ~~ { A } )
6		simpr	|- ( ( A e. V /\ -. A e. A ) -> -. A e. A )
7		disjsn	|- ( ( A i^i { A } ) = (/) <-> -. A e. A )
8	6 7	sylibr	|- ( ( A e. V /\ -. A e. A ) -> ( A i^i { A } ) = (/) )
9		djuenun	|- ( ( A ~~ A /\ 1o ~~ { A } /\ ( A i^i { A } ) = (/) ) -> ( A |_| 1o ) ~~ ( A u. { A } ) )
10	2 5 8 9	syl3anc	|- ( ( A e. V /\ -. A e. A ) -> ( A |_| 1o ) ~~ ( A u. { A } ) )
11		df-suc	|- suc A = ( A u. { A } )
12	10 11	breqtrrdi	|- ( ( A e. V /\ -. A e. A ) -> ( A |_| 1o ) ~~ suc A )