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Theorem eldju2ndr 6671
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.)
Assertion
Ref Expression
eldju2ndr  |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =/=  (/) )  ->  ( 2nd `  X )  e.  B
)

Proof of Theorem eldju2ndr
StepHypRef Expression
1 df-dju 6638 . . . . 5  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
21eleq2i 2149 . . . 4  |-  ( X  e.  ( A B )  <-> 
X  e.  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
3 elun 3125 . . . 4  |-  ( X  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) )  <->  ( X  e.  ( { (/) }  X.  A )  \/  X  e.  ( { 1o }  X.  B ) ) )
42, 3bitri 182 . . 3  |-  ( X  e.  ( A B )  <-> 
( X  e.  ( { (/) }  X.  A
)  \/  X  e.  ( { 1o }  X.  B ) ) )
5 elxp6 5875 . . . . 5  |-  ( X  e.  ( { (/) }  X.  A )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { (/) }  /\  ( 2nd `  X
)  e.  A ) ) )
6 elsni 3440 . . . . . . 7  |-  ( ( 1st `  X )  e.  { (/) }  ->  ( 1st `  X )  =  (/) )
7 eqneqall 2259 . . . . . . 7  |-  ( ( 1st `  X )  =  (/)  ->  ( ( 1st `  X )  =/=  (/)  ->  ( 2nd `  X )  e.  B
) )
86, 7syl 14 . . . . . 6  |-  ( ( 1st `  X )  e.  { (/) }  ->  ( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
98ad2antrl 474 . . . . 5  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ (/) }  /\  ( 2nd `  X )  e.  A ) )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
105, 9sylbi 119 . . . 4  |-  ( X  e.  ( { (/) }  X.  A )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
11 elxp6 5875 . . . . 5  |-  ( X  e.  ( { 1o }  X.  B )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { 1o }  /\  ( 2nd `  X
)  e.  B ) ) )
12 simprr 499 . . . . . 6  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ 1o }  /\  ( 2nd `  X )  e.  B ) )  ->  ( 2nd `  X
)  e.  B )
1312a1d 22 . . . . 5  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ 1o }  /\  ( 2nd `  X )  e.  B ) )  ->  ( ( 1st `  X )  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
1411, 13sylbi 119 . . . 4  |-  ( X  e.  ( { 1o }  X.  B )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
1510, 14jaoi 669 . . 3  |-  ( ( X  e.  ( {
(/) }  X.  A
)  \/  X  e.  ( { 1o }  X.  B ) )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
164, 15sylbi 119 . 2  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
1716imp 122 1  |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =/=  (/) )  ->  ( 2nd `  X )  e.  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434    =/= wne 2249    u. cun 2982   (/)c0 3269   {csn 3422   <.cop 3425    X. cxp 4399   ` cfv 4969   1stc1st 5844   2ndc2nd 5845   1oc1o 6106   ⊔ cdju 6637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fv 4977  df-1st 5846  df-2nd 5847  df-dju 6638
This theorem is referenced by:  updjudhf  6677
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