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Theorem eldju2ndl 6742
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.)
Assertion
Ref Expression
eldju2ndl  |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =  (/) )  ->  ( 2nd `  X )  e.  A
)

Proof of Theorem eldju2ndl
StepHypRef Expression
1 df-dju 6710 . . . . 5  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
21eleq2i 2154 . . . 4  |-  ( X  e.  ( A B )  <-> 
X  e.  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
3 elun 3139 . . . 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 5922 . . . . 5  |-  ( X  e.  ( { (/) }  X.  A )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { (/) }  /\  ( 2nd `  X
)  e.  A ) ) )
6 simprr 499 . . . . . 6  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ (/) }  /\  ( 2nd `  X )  e.  A ) )  -> 
( 2nd `  X
)  e.  A )
76a1d 22 . . . . 5  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ (/) }  /\  ( 2nd `  X )  e.  A ) )  -> 
( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
85, 7sylbi 119 . . . 4  |-  ( X  e.  ( { (/) }  X.  A )  -> 
( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
9 elxp6 5922 . . . . 5  |-  ( X  e.  ( { 1o }  X.  B )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { 1o }  /\  ( 2nd `  X
)  e.  B ) ) )
10 elsni 3459 . . . . . . 7  |-  ( ( 1st `  X )  e.  { 1o }  ->  ( 1st `  X
)  =  1o )
11 1n0 6179 . . . . . . . 8  |-  1o  =/=  (/)
12 neeq1 2268 . . . . . . . 8  |-  ( ( 1st `  X )  =  1o  ->  (
( 1st `  X
)  =/=  (/)  <->  1o  =/=  (/) ) )
1311, 12mpbiri 166 . . . . . . 7  |-  ( ( 1st `  X )  =  1o  ->  ( 1st `  X )  =/=  (/) )
14 eqneqall 2265 . . . . . . . 8  |-  ( ( 1st `  X )  =  (/)  ->  ( ( 1st `  X )  =/=  (/)  ->  ( 2nd `  X )  e.  A
) )
1514com12 30 . . . . . . 7  |-  ( ( 1st `  X )  =/=  (/)  ->  ( ( 1st `  X )  =  (/)  ->  ( 2nd `  X
)  e.  A ) )
1610, 13, 153syl 17 . . . . . 6  |-  ( ( 1st `  X )  e.  { 1o }  ->  ( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
1716ad2antrl 474 . . . . 5  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ 1o }  /\  ( 2nd `  X )  e.  B ) )  ->  ( ( 1st `  X )  =  (/)  ->  ( 2nd `  X
)  e.  A ) )
189, 17sylbi 119 . . . 4  |-  ( X  e.  ( { 1o }  X.  B )  -> 
( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
198, 18jaoi 671 . . 3  |-  ( ( X  e.  ( {
(/) }  X.  A
)  \/  X  e.  ( { 1o }  X.  B ) )  -> 
( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
204, 19sylbi 119 . 2  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =  (/)  ->  ( 2nd `  X
)  e.  A ) )
2120imp 122 1  |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =  (/) )  ->  ( 2nd `  X )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438    =/= wne 2255    u. cun 2995   (/)c0 3284   {csn 3441   <.cop 3444    X. cxp 4426   ` cfv 5002   1stc1st 5891   2ndc2nd 5892   1oc1o 6156   ⊔ cdju 6709
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710
This theorem is referenced by:  updjudhf  6749
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