Theorem eldju2ndr 7403

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 7368	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2	1	eleq2i 2305	. . . 4 |- ( X e. ( A ⊔ B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3		elun 3370	. . . 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 6393	. . . . 5 |- ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6		elsni 3723	. . . . . . 7 |- ( ( 1st ` X ) e. { (/) } -> ( 1st ` X ) = (/) )
7		eqneqall 2430	. . . . . . 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 494	. . . . 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 6393	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
12		simprr 537	. . . . . 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 728	. . 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 720   = wceq 1402   e. wcel 2209   =/= wne 2420   u. cun 3218   (/)c0 3520   {csn 3705   <.cop 3708   X. cxp 4767   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   1oc1o 6670   ⊔ cdju 7367

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 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-1st 6364  df-2nd 6365  df-dju 7368

This theorem is referenced by:  updjudhf  7409