Theorem eldju2ndl 7405

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 7371	. . . . 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 6396	. . . . 5 |- ( X e. ( { (/) } X. A ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { (/) } /\ ( 2nd ` X ) e. A ) ) )
6		simprr 537	. . . . . 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 6396	. . . . 5 |- ( X e. ( { 1o } X. B ) <-> ( X = <. ( 1st ` X ) , ( 2nd ` X ) >. /\ ( ( 1st ` X ) e. { 1o } /\ ( 2nd ` X ) e. B ) ) )
10		elsni 3726	. . . . . . 7 |- ( ( 1st ` X ) e. { 1o } -> ( 1st ` X ) = 1o )
11		1n0 6698	. . . . . . . 8 |- 1o =/= (/)
12		neeq1 2433	. . . . . . . 8 |- ( ( 1st ` X ) = 1o -> ( ( 1st ` X ) =/= (/) <-> 1o =/= (/) ) )
13	11, 12	mpbiri 168	. . . . . . 7 |- ( ( 1st ` X ) = 1o -> ( 1st ` X ) =/= (/) )
14		eqneqall 2430	. . . . . . . 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 494	. . . . 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 728	. . 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 720   = wceq 1402   e. wcel 2209   =/= wne 2420   u. cun 3218   (/)c0 3520   {csn 3708   <.cop 3711   X. cxp 4770   ` cfv 5375   1stc1st 6365   2ndc2nd 6366   1oc1o 6673   ⊔ cdju 7370

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 623  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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-1st 6367  df-2nd 6368  df-1o 6680  df-dju 7371

This theorem is referenced by:  updjudhf  7412  subctctexmid  16947