Theorem eldju2ndl 7006

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 6972	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2	1	eleq2i 2224	. . . 4 |- ( X e. ( A ⊔ B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3		elun 3248	. . . 4 |- ( X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) )
4	2, 3	bitri 183	. . 3 |- ( X e. ( A ⊔ B ) <-> ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) )
5		elxp6 6111	. . . . 5 |- ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6		simprr 522	. . . . . 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 120	. . . 4 |- ( X e. ( { (/) } X. A ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
9		elxp6 6111	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
10		elsni 3578	. . . . . . 7 |- ( ( 1st ` X ) e. { 1o } -> ( 1st ` X ) = 1o )
11		1n0 6373	. . . . . . . 8 |- 1o =/= (/)
12		neeq1 2340	. . . . . . . 8 |- ( ( 1st ` X ) = 1o -> ( ( 1st ` X ) =/= (/) <-> 1o =/= (/) ) )
13	11, 12	mpbiri 167	. . . . . . 7 |- ( ( 1st ` X ) = 1o -> ( 1st ` X ) =/= (/) )
14		eqneqall 2337	. . . . . . . 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 482	. . . . 5 |- ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
18	9, 17	sylbi 120	. . . 4 |- ( X e. ( { 1o } X. B ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
19	8, 18	jaoi 706	. . 3 |- ( ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
20	4, 19	sylbi 120	. 2 |- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) -> ( 2nd ` X ) e. A ) )
21	20	imp 123	1 |- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) = (/) ) -> ( 2nd ` X ) e. A )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1335   e. wcel 2128   =/= wne 2327   u. cun 3100   (/)c0 3394   {csn 3560   <.cop 3563   X. cxp 4581   ` cfv 5167   1stc1st 6080   2ndc2nd 6081   1oc1o 6350   ⊔ cdju 6971

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fv 5175  df-1st 6082  df-2nd 6083  df-1o 6357  df-dju 6972

This theorem is referenced by:  updjudhf  7013  subctctexmid  13534