Theorem eldju1st 7089

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 7088	. 2 |- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
2		ssel2 3165	. . 3 |- ( ( ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) ) /\ X e. ( A ⊔ B ) ) -> X e. ( { (/) , 1o } X. ( A u. B ) ) )
3		xp1st 6184	. . 3 |- ( X e. ( { (/) , 1o } X. ( A u. B ) ) -> ( 1st ` X ) e. { (/) , 1o } )
4		elpri 3630	. . 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 424	1 |- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2160   u. cun 3142   C_ wss 3144   (/)c0 3437   {cpr 3608   X. cxp 4639   ` cfv 5231   1stc1st 6157   1oc1o 6428   ⊔ cdju 7055

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-1st 6159  df-2nd 6160  df-1o 6435  df-dju 7056  df-inl 7065  df-inr 7066

This theorem is referenced by:  updjudhf  7097  subctctexmid  15154