Theorem eldju1st 7411

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

Step	Hyp	Ref	Expression
1		djuss 7410	. 2 |- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
2		ssel2 3243	. . 3 |- ( ( ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) ) /\ X e. ( A ⊔ B ) ) -> X e. ( { (/) , 1o } X. ( A u. B ) ) )
3		xp1st 6399	. . 3 |- ( X e. ( { (/) , 1o } X. ( A u. B ) ) -> ( 1st ` X ) e. { (/) , 1o } )
4		elpri 3732	. . 3 |- ( ( 1st ` X ) e. { (/) , 1o } -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )
5	2, 3, 4	3syl 17	. 2 |- ( ( ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) ) /\ X e. ( A ⊔ B ) ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )
6	1, 5	mpan 428	1 |- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   u. cun 3218   C_ wss 3220   (/)c0 3520   {cpr 3710   X. cxp 4772   ` cfv 5377   1stc1st 6372   1oc1o 6680   ⊔ cdju 7377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375  df-1o 6687  df-dju 7378  df-inl 7387  df-inr 7388

This theorem is used by:  updjudhf  7419  subctctexmid  17042