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Theorem eldju2ndl 6923
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 6889 . . . . 5  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
21eleq2i 2182 . . . 4  |-  ( X  e.  ( A B )  <-> 
X  e.  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
3 elun 3185 . . . 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 6033 . . . . 5  |-  ( X  e.  ( { (/) }  X.  A )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { (/) }  /\  ( 2nd `  X
)  e.  A ) ) )
6 simprr 504 . . . . . 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 120 . . . 4  |-  ( X  e.  ( { (/) }  X.  A )  -> 
( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
9 elxp6 6033 . . . . 5  |-  ( X  e.  ( { 1o }  X.  B )  <->  ( X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  /\  (
( 1st `  X
)  e.  { 1o }  /\  ( 2nd `  X
)  e.  B ) ) )
10 elsni 3513 . . . . . . 7  |-  ( ( 1st `  X )  e.  { 1o }  ->  ( 1st `  X
)  =  1o )
11 1n0 6295 . . . . . . . 8  |-  1o  =/=  (/)
12 neeq1 2296 . . . . . . . 8  |-  ( ( 1st `  X )  =  1o  ->  (
( 1st `  X
)  =/=  (/)  <->  1o  =/=  (/) ) )
1311, 12mpbiri 167 . . . . . . 7  |-  ( ( 1st `  X )  =  1o  ->  ( 1st `  X )  =/=  (/) )
14 eqneqall 2293 . . . . . . . 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 479 . . . . 5  |-  ( ( X  =  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  /\  ( ( 1st `  X )  e. 
{ 1o }  /\  ( 2nd `  X )  e.  B ) )  ->  ( ( 1st `  X )  =  (/)  ->  ( 2nd `  X
)  e.  A ) )
189, 17sylbi 120 . . . 4  |-  ( X  e.  ( { 1o }  X.  B )  -> 
( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
198, 18jaoi 688 . . 3  |-  ( ( X  e.  ( {
(/) }  X.  A
)  \/  X  e.  ( { 1o }  X.  B ) )  -> 
( ( 1st `  X
)  =  (/)  ->  ( 2nd `  X )  e.  A ) )
204, 19sylbi 120 . 2  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =  (/)  ->  ( 2nd `  X
)  e.  A ) )
2120imp 123 1  |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =  (/) )  ->  ( 2nd `  X )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314    e. wcel 1463    =/= wne 2283    u. cun 3037   (/)c0 3331   {csn 3495   <.cop 3498    X. cxp 4505   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   1oc1o 6272   ⊔ cdju 6888
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889
This theorem is referenced by:  updjudhf  6930  subctctexmid  12998
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