Theorem eldju2ndl 7133

Description: The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)

Assertion

Ref	Expression
eldju2ndl	|- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) = (/) ) -> ( 2nd ` X ) e. A )

Proof of Theorem eldju2ndl

Step	Hyp	Ref	Expression
1		df-dju 7099	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2	1	eleq2i 2260	. . . 4 |- ( X e. ( A ⊔ B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3		elun 3301	. . . 4 |- ( X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) )
4	2, 3	bitri 184	. . 3 |- ( X e. ( A ⊔ B ) <-> ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) )
5		elxp6 6224	. . . . 5 |- ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6		simprr 531	. . . . . 6 |- ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) -> ( 2nd ` X ) e. A )
7	6	a1d 22	. . . . 5 |- ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
8	5, 7	sylbi 121	. . . 4 |- ( X e. ( { (/) } X. A ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
9		elxp6 6224	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
10		elsni 3637	. . . . . . 7 |- ( ( 1st ` X ) e. { 1o } -> ( 1st ` X ) = 1o )
11		1n0 6487	. . . . . . . 8 |- 1o =/= (/)
12		neeq1 2377	. . . . . . . 8 |- ( ( 1st ` X ) = 1o -> ( ( 1st ` X ) =/= (/) <-> 1o =/= (/) ) )
13	11, 12	mpbiri 168	. . . . . . 7 |- ( ( 1st ` X ) = 1o -> ( 1st ` X ) =/= (/) )
14		eqneqall 2374	. . . . . . . 8 |- ( ( 1st ` X ) = (/) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. A ) )
15	14	com12 30	. . . . . . 7 |- ( ( 1st ` X ) =/= (/) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
16	10, 13, 15	3syl 17	. . . . . 6 |- ( ( 1st ` X ) e. { 1o } -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
17	16	ad2antrl 490	. . . . 5 |- ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
18	9, 17	sylbi 121	. . . 4 |- ( X e. ( { 1o } X. B ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
19	8, 18	jaoi 717	. . 3 |- ( ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
20	4, 19	sylbi 121	. 2 |- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
21	20	imp 124	1 |- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) = (/) ) -> ( 2nd ` X ) e. A )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   =/= wne 2364   u. cun 3152   (/)c0 3447   {csn 3619   <.cop 3622   X. cxp 4658   ` cfv 5255   1stc1st 6193   2ndc2nd 6194   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-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  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-id 4325  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-dju 7099

This theorem is referenced by:  updjudhf  7140  subctctexmid  15561