Metamath Proof Explorer

Theorem eldju2ndl

Description: The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022)

Ref Expression
Assertion eldju2ndl
`|- ( ( X e. ( A |_| B ) /\ ( 1st ` X ) = (/) ) -> ( 2nd ` X ) e. A )`

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 simprr
` |-  ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) -> ( 2nd ` X ) e. A )`
7 6 a1d
` |-  ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
8 5 7 sylbi
` |-  ( X e. ( { (/) } X. A ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
9 elxp6
` |-  ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )`
10 elsni
` |-  ( ( 1st ` X ) e. { 1o } -> ( 1st ` X ) = 1o )`
11 1n0
` |-  1o =/= (/)`
12 neeq1
` |-  ( ( 1st ` X ) = 1o -> ( ( 1st ` X ) =/= (/) <-> 1o =/= (/) ) )`
13 11 12 mpbiri
` |-  ( ( 1st ` X ) = 1o -> ( 1st ` X ) =/= (/) )`
14 eqneqall
` |-  ( ( 1st ` X ) = (/) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. A ) )`
15 14 com12
` |-  ( ( 1st ` X ) =/= (/) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
16 10 13 15 3syl
` |-  ( ( 1st ` X ) e. { 1o } -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
` |-  ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
` |-  ( X e. ( { 1o } X. B ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
` |-  ( ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
` |-  ( X e. ( A |_| B ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )`
` |-  ( ( X e. ( A |_| B ) /\ ( 1st ` X ) = (/) ) -> ( 2nd ` X ) e. A )`