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Theorem eldju1st 6741
Description: The first component of an element of a disjoint union is either  (/) or  1o. (Contributed by AV, 26-Jun-2022.)
Assertion
Ref Expression
eldju1st  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =  (/)  \/  ( 1st `  X
)  =  1o ) )

Proof of Theorem eldju1st
StepHypRef Expression
1 djuss 6740 . 2  |-  ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B
) )
2 ssel2 3018 . . 3  |-  ( ( ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B ) )  /\  X  e.  ( A B ) )  ->  X  e.  ( { (/) ,  1o }  X.  ( A  u.  B ) ) )
3 xp1st 5918 . . 3  |-  ( X  e.  ( { (/) ,  1o }  X.  ( A  u.  B )
)  ->  ( 1st `  X )  e.  { (/)
,  1o } )
4 elpri 3464 . . 3  |-  ( ( 1st `  X )  e.  { (/) ,  1o }  ->  ( ( 1st `  X )  =  (/)  \/  ( 1st `  X
)  =  1o ) )
52, 3, 43syl 17 . 2  |-  ( ( ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B ) )  /\  X  e.  ( A B ) )  ->  (
( 1st `  X
)  =  (/)  \/  ( 1st `  X )  =  1o ) )
61, 5mpan 415 1  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =  (/)  \/  ( 1st `  X
)  =  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438    u. cun 2995    C_ wss 2997   (/)c0 3284   {cpr 3442    X. cxp 4426   ` cfv 5002   1stc1st 5891   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-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  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-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  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-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710  df-inl 6718  df-inr 6719
This theorem is referenced by:  updjudhf  6749
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