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Metamath Proof Explorer

Theorem eldju2ndr

Description: The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022)

Ref Expression
Assertion eldju2ndr 
|- ( ( X e. ( A |_| B ) /\ ( 1st ` X ) =/= (/) ) -> ( 2nd ` X ) e. B )

Proof

Step Hyp Ref Expression
1 df-dju 
 |-  ( A |_| B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2 1 eleq2i 
 |-  ( X e. ( A |_| B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3 elun 
 |-  ( X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) )
4 2 3 bitri 
 |-  ( X e. ( A |_| B ) <-> ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) )
5 elxp6 
 |-  ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6 elsni 
 |-  ( ( 1st ` X ) e. { (/) } -> ( 1st ` X ) = (/) )
7 eqneqall 
 |-  ( ( 1st ` X ) = (/) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
8 6 7 syl 
 |-  ( ( 1st ` X ) e. { (/) } -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
9 8 ad2antrl 
 |-  ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
10 5 9 sylbi 
 |-  ( X e. ( { (/) } X. A ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
11 elxp6 
 |-  ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
12 simprr 
 |-  ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) -> ( 2nd ` X ) e. B )
13 12 a1d 
 |-  ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
14 11 13 sylbi 
 |-  ( X e. ( { 1o } X. B ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
15 10 14 jaoi 
 |-  ( ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
16 4 15 sylbi 
 |-  ( X e. ( A |_| B ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
17 16 imp 
 |-  ( ( X e. ( A |_| B ) /\ ( 1st ` X ) =/= (/) ) -> ( 2nd ` X ) e. B )