Theorem eldju1st 6956

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 6955	. 2 |- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
2		ssel2 3092	. . 3 |- ( ( ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) ) /\ X e. ( A ⊔ B ) ) -> X e. ( { (/) , 1o } X. ( A u. B ) ) )
3		xp1st 6063	. . 3 |- ( X e. ( { (/) , 1o } X. ( A u. B ) ) -> ( 1st ` X ) e. { (/) , 1o } )
4		elpri 3550	. . 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 420	1 |- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   u. cun 3069   C_ wss 3071   (/)c0 3363   {cpr 3528   X. cxp 4537   ` cfv 5123   1stc1st 6036   1oc1o 6306   ⊔ cdju 6922

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933

This theorem is referenced by:  updjudhf  6964  subctctexmid  13196