Theorem eldju2ndr 7201

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 7166	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2	1	eleq2i 2274	. . . 4 |- ( X e. ( A ⊔ B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3		elun 3322	. . . 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 6278	. . . . 5 |- ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6		elsni 3661	. . . . . . 7 |- ( ( 1st ` X ) e. { (/) } -> ( 1st ` X ) = (/) )
7		eqneqall 2388	. . . . . . 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 6278	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
12		simprr 531	. . . . . 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 718	. . 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 710   = wceq 1373   e. wcel 2178   =/= wne 2378   u. cun 3172   (/)c0 3468   {csn 3643   <.cop 3646   X. cxp 4691   ` cfv 5290   1stc1st 6247   2ndc2nd 6248   1oc1o 6518   ⊔ cdju 7165

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 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fv 5298  df-1st 6249  df-2nd 6250  df-dju 7166

This theorem is referenced by:  updjudhf  7207