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Metamath Proof Explorer

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 )