Theorem eldju2ndr 7364

Description: The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)

Assertion

Ref	Expression
eldju2ndr	|- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) =/= (/) ) -> ( 2nd ` X ) e. B )

Proof of Theorem eldju2ndr

Step	Hyp	Ref	Expression
1		df-dju 7329	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2	1	eleq2i 2299	. . . 4 |- ( X e. ( A ⊔ B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3		elun 3360	. . . 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 6363	. . . . 5 |- ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6		elsni 3707	. . . . . . 7 |- ( ( 1st ` X ) e. { (/) } -> ( 1st ` X ) = (/) )
7		eqneqall 2422	. . . . . . 7 |- ( ( 1st ` X ) = (/) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
8	6, 7	syl 14	. . . . . 6 |- ( ( 1st ` X ) e. { (/) } -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
9	8	ad2antrl 490	. . . . 5 |- ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
10	5, 9	sylbi 121	. . . 4 |- ( X e. ( { (/) } X. A ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
11		elxp6 6363	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
12		simprr 533	. . . . . 6 |- ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) -> ( 2nd ` X ) e. B )
13	12	a1d 22	. . . . 5 |- ( ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
14	11, 13	sylbi 121	. . . 4 |- ( X e. ( { 1o } X. B ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
15	10, 14	jaoi 724	. . 3 |- ( ( X e. ( { (/) } X. A ) \/ X e. ( { 1o } X. B ) ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
16	4, 15	sylbi 121	. 2 |- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) =/= (/) -> ( 2nd ` X ) e. B ) )
17	16	imp 124	1 |- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) =/= (/) ) -> ( 2nd ` X ) e. B )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   =/= wne 2412   u. cun 3209   (/)c0 3508   {csn 3689   <.cop 3692   X. cxp 4747   ` cfv 5352   1stc1st 6332   2ndc2nd 6333   1oc1o 6640   ⊔ cdju 7328

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-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-1st 6334  df-2nd 6335  df-dju 7329

This theorem is referenced by:  updjudhf  7370