Theorem eldju2ndl 7138

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 7104	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2	1	eleq2i 2263	. . . 4 |- ( X e. ( A ⊔ B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3		elun 3304	. . . 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 6227	. . . . 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 6227	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
10		elsni 3640	. . . . . . 7 |- ( ( 1st ` X ) e. { 1o } -> ( 1st ` X ) = 1o )
11		1n0 6490	. . . . . . . 8 |- 1o =/= (/)
12		neeq1 2380	. . . . . . . 8 |- ( ( 1st ` X ) = 1o -> ( ( 1st ` X ) =/= (/) <-> 1o =/= (/) ) )
13	11, 12	mpbiri 168	. . . . . . 7 |- ( ( 1st ` X ) = 1o -> ( 1st ` X ) =/= (/) )
14		eqneqall 2377	. . . . . . . 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 2167   =/= wne 2367   u. cun 3155   (/)c0 3450   {csn 3622   <.cop 3625   X. cxp 4661   ` cfv 5258   1stc1st 6196   2ndc2nd 6197   1oc1o 6467   ⊔ cdju 7103

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104

This theorem is referenced by:  updjudhf  7145  subctctexmid  15645