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Theorem eldju1st 6960
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 6959 . 2  |-  ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B
) )
2 ssel2 3093 . . 3  |-  ( ( ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B ) )  /\  X  e.  ( A B ) )  ->  X  e.  ( { (/) ,  1o }  X.  ( A  u.  B ) ) )
3 xp1st 6067 . . 3  |-  ( X  e.  ( { (/) ,  1o }  X.  ( A  u.  B )
)  ->  ( 1st `  X )  e.  { (/)
,  1o } )
4 elpri 3551 . . 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 421 1  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =  (/)  \/  ( 1st `  X
)  =  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1332    e. wcel 1481    u. cun 3070    C_ wss 3072   (/)c0 3364   {cpr 3529    X. cxp 4541   ` cfv 5127   1stc1st 6040   1oc1o 6310   ⊔ cdju 6926
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 604  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 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359
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-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-1st 6042  df-2nd 6043  df-1o 6317  df-dju 6927  df-inl 6936  df-inr 6937
This theorem is referenced by:  updjudhf  6968  subctctexmid  13352
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