Theorem eldju2ndl 7331

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 7297	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2	1	eleq2i 2298	. . . 4 |- ( X e. ( A ⊔ B ) <-> X e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
3		elun 3350	. . . 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 6341	. . . . 5 |- ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6		simprr 533	. . . . . 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 6341	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
10		elsni 3691	. . . . . . 7 |- ( ( 1st ` X ) e. { 1o } -> ( 1st ` X ) = 1o )
11		1n0 6643	. . . . . . . 8 |- 1o =/= (/)
12		neeq1 2416	. . . . . . . 8 |- ( ( 1st ` X ) = 1o -> ( ( 1st ` X ) =/= (/) <-> 1o =/= (/) ) )
13	11, 12	mpbiri 168	. . . . . . 7 |- ( ( 1st ` X ) = 1o -> ( 1st ` X ) =/= (/) )
14		eqneqall 2413	. . . . . . . 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 724	. . 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 716   = wceq 1398   e. wcel 2202   =/= wne 2403   u. cun 3199   (/)c0 3496   {csn 3673   <.cop 3676   X. cxp 4729   ` cfv 5333   1stc1st 6310   2ndc2nd 6311   1oc1o 6618   ⊔ cdju 7296

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 619  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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-dju 7297

This theorem is referenced by:  updjudhf  7338  subctctexmid  16722