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Theorem eldju2ndr 6965
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 6930 . . . . 5  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
21eleq2i 2207 . . . 4  |-  ( X  e.  ( A B )  <-> 
X  e.  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
3 elun 3221 . . . 4  |-  ( X  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) )  <->  ( X  e.  ( { (/) }  X.  A )  \/  X  e.  ( { 1o }  X.  B ) ) )
42, 3bitri 183 . . 3  |-  ( X  e.  ( A B )  <-> 
( X  e.  ( { (/) }  X.  A
)  \/  X  e.  ( { 1o }  X.  B ) ) )
5 elxp6 6074 . . . . 5  |-  ( X  e.  ( { (/) }  X.  A )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { (/) }  /\  ( 2nd `  X
)  e.  A ) ) )
6 elsni 3549 . . . . . . 7  |-  ( ( 1st `  X )  e.  { (/) }  ->  ( 1st `  X )  =  (/) )
7 eqneqall 2319 . . . . . . 7  |-  ( ( 1st `  X )  =  (/)  ->  ( ( 1st `  X )  =/=  (/)  ->  ( 2nd `  X )  e.  B
) )
86, 7syl 14 . . . . . 6  |-  ( ( 1st `  X )  e.  { (/) }  ->  ( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
98ad2antrl 482 . . . . 5  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ (/) }  /\  ( 2nd `  X )  e.  A ) )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
105, 9sylbi 120 . . . 4  |-  ( X  e.  ( { (/) }  X.  A )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
11 elxp6 6074 . . . . 5  |-  ( X  e.  ( { 1o }  X.  B )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { 1o }  /\  ( 2nd `  X
)  e.  B ) ) )
12 simprr 522 . . . . . 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 120 . . . 4  |-  ( X  e.  ( { 1o }  X.  B )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
1510, 14jaoi 706 . . 3  |-  ( ( X  e.  ( {
(/) }  X.  A
)  \/  X  e.  ( { 1o }  X.  B ) )  -> 
( ( 1st `  X
)  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
164, 15sylbi 120 . 2  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =/=  (/)  ->  ( 2nd `  X )  e.  B ) )
1716imp 123 1  |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =/=  (/) )  ->  ( 2nd `  X )  e.  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1332    e. wcel 1481    =/= wne 2309    u. cun 3073   (/)c0 3367   {csn 3531   <.cop 3534    X. cxp 4544   ` cfv 5130   1stc1st 6043   2ndc2nd 6044   1oc1o 6313   ⊔ cdju 6929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fv 5138  df-1st 6045  df-2nd 6046  df-dju 6930
This theorem is referenced by:  updjudhf  6971
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