Theorem eldju1st 7132

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 7131	. 2 |- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
2		ssel2 3175	. . 3 |- ( ( ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) ) /\ X e. ( A ⊔ B ) ) -> X e. ( { (/) , 1o } X. ( A u. B ) ) )
3		xp1st 6220	. . 3 |- ( X e. ( { (/) , 1o } X. ( A u. B ) ) -> ( 1st ` X ) e. { (/) , 1o } )
4		elpri 3642	. . 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 2164   u. cun 3152   C_ wss 3154   (/)c0 3447   {cpr 3620   X. cxp 4658   ` cfv 5255   1stc1st 6193   1oc1o 6464   ⊔ cdju 7098

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-dju 7099  df-inl 7108  df-inr 7109

This theorem is referenced by:  updjudhf  7140  subctctexmid  15561