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Theorem eldju1st 7083
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 7082 . 2  |-  ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B
) )
2 ssel2 3162 . . 3  |-  ( ( ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B ) )  /\  X  e.  ( A B ) )  ->  X  e.  ( { (/) ,  1o }  X.  ( A  u.  B ) ) )
3 xp1st 6179 . . 3  |-  ( X  e.  ( { (/) ,  1o }  X.  ( A  u.  B )
)  ->  ( 1st `  X )  e.  { (/)
,  1o } )
4 elpri 3627 . . 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 424 1  |-  ( X  e.  ( A B )  ->  ( ( 1st `  X )  =  (/)  \/  ( 1st `  X
)  =  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709    = wceq 1363    e. wcel 2158    u. cun 3139    C_ wss 3141   (/)c0 3434   {cpr 3605    X. cxp 4636   ` cfv 5228   1stc1st 6152   1oc1o 6423   ⊔ cdju 7049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6154  df-2nd 6155  df-1o 6430  df-dju 7050  df-inl 7059  df-inr 7060
This theorem is referenced by:  updjudhf  7091  subctctexmid  14991
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