Theorem idjudju 17252

Description: An indexed disjoint union over two sets equals the disjoint union of those two sets. This connects our notation for an indexed disjoint union, U_ x e. C ( { x } X. F ) which uses df-iun 4014, with our notation for the disjoint union of two sets (df-dju 7384). (Contributed by Jim Kingdon, 2-Oct-2026.)

Hypothesis

Ref	Expression
idjudju.f	|- F = if ( x = (/) , A , B )

Assertion

Ref	Expression
idjudju	|- U_ x e. 2o ( { x } X. F ) = ( A ⊔ B )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   F( x)

Proof of Theorem idjudju

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliunxp 4921	. . . 4 |- ( y e. U_ x e. 2o ( { x } X. F ) <-> E. x E. z ( y = <. x , z >. /\ ( x e. 2o /\ z e. F ) ) )
2		simpl 109	. . . . . 6 |- ( ( y = <. x , z >. /\ ( x e. 2o /\ z e. F ) ) -> y = <. x , z >. )
3		anass 405	. . . . . . . 8 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x e. 2o ) <-> ( y = <. x , z >. /\ ( z e. F /\ x e. 2o ) ) )
4		ancom 266	. . . . . . . . 9 |- ( ( z e. F /\ x e. 2o ) <-> ( x e. 2o /\ z e. F ) )
5	4	anbi2i 461	. . . . . . . 8 |- ( ( y = <. x , z >. /\ ( z e. F /\ x e. 2o ) ) <-> ( y = <. x , z >. /\ ( x e. 2o /\ z e. F ) ) )
6	3, 5	bitri 184	. . . . . . 7 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x e. 2o ) <-> ( y = <. x , z >. /\ ( x e. 2o /\ z e. F ) ) )
7		elpri 3732	. . . . . . . . 9 |- ( x e. { (/) , 1o } -> ( x = (/) \/ x = 1o ) )
8		df2o3 6706	. . . . . . . . 9 |- 2o = { (/) , 1o }
9	7, 8	eleq2s 2333	. . . . . . . 8 |- ( x e. 2o -> ( x = (/) \/ x = 1o ) )
10		ssun1 3392	. . . . . . . . . . 11 |- ( { (/) } X. A ) C_ ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
11		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> x = (/) )
12		0ex 4262	. . . . . . . . . . . . . 14 |- (/) e. _V
13	12	snid 3740	. . . . . . . . . . . . 13 |- (/) e. { (/) }
14	11, 13	eqeltrdi 2329	. . . . . . . . . . . 12 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> x e. { (/) } )
15	11	notnotd 639	. . . . . . . . . . . . . . 15 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> -. -. x = (/) )
16	15	intnanrd 944	. . . . . . . . . . . . . 14 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> -. ( -. x = (/) /\ z e. B ) )
17		idjudju.f	. . . . . . . . . . . . . . . . . 18 |- F = if ( x = (/) , A , B )
18	17	eleq2i 2305	. . . . . . . . . . . . . . . . 17 |- ( z e. F <-> z e. if ( x = (/) , A , B ) )
19		elif 3652	. . . . . . . . . . . . . . . . 17 |- ( z e. if ( x = (/) , A , B ) <-> ( ( x = (/) /\ z e. A ) \/ ( -. x = (/) /\ z e. B ) ) )
20	18, 19	bitri 184	. . . . . . . . . . . . . . . 16 |- ( z e. F <-> ( ( x = (/) /\ z e. A ) \/ ( -. x = (/) /\ z e. B ) ) )
21	20	bilani 387	. . . . . . . . . . . . . . 15 |- ( ( y = <. x , z >. /\ z e. F ) -> ( ( x = (/) /\ z e. A ) \/ ( -. x = (/) /\ z e. B ) ) )
22	21	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> ( ( x = (/) /\ z e. A ) \/ ( -. x = (/) /\ z e. B ) ) )
23	16, 22	ecased 1390	. . . . . . . . . . . . 13 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> ( x = (/) /\ z e. A ) )
24	23	simprd 114	. . . . . . . . . . . 12 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> z e. A )
25	14, 24	opelxpd 4809	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> <. x , z >. e. ( { (/) } X. A ) )
26	10, 25	sselid 3246	. . . . . . . . . 10 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> <. x , z >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
27		df-dju 7384	. . . . . . . . . 10 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
28	26, 27	eleqtrrdi 2332	. . . . . . . . 9 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = (/) ) -> <. x , z >. e. ( A ⊔ B ) )
29		ssun2 3393	. . . . . . . . . . 11 |- ( { 1o } X. B ) C_ ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
30		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> x = 1o )
31		1oex 6699	. . . . . . . . . . . . . 14 |- 1o e. _V
32	31	snid 3740	. . . . . . . . . . . . 13 |- 1o e. { 1o }
33	30, 32	eqeltrdi 2329	. . . . . . . . . . . 12 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> x e. { 1o } )
34		1n0 6709	. . . . . . . . . . . . . . . . . 18 |- 1o =/= (/)
35	34	neii 2422	. . . . . . . . . . . . . . . . 17 |- -. 1o = (/)
36		eqtr2 2257	. . . . . . . . . . . . . . . . . 18 |- ( ( x = 1o /\ x = (/) ) -> 1o = (/) )
37	36	ex 115	. . . . . . . . . . . . . . . . 17 |- ( x = 1o -> ( x = (/) -> 1o = (/) ) )
38	35, 37	mtoi 674	. . . . . . . . . . . . . . . 16 |- ( x = 1o -> -. x = (/) )
39	38	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> -. x = (/) )
40	39	intnanrd 944	. . . . . . . . . . . . . 14 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> -. ( x = (/) /\ z e. A ) )
41	21	orcomd 741	. . . . . . . . . . . . . . 15 |- ( ( y = <. x , z >. /\ z e. F ) -> ( ( -. x = (/) /\ z e. B ) \/ ( x = (/) /\ z e. A ) ) )
42	41	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> ( ( -. x = (/) /\ z e. B ) \/ ( x = (/) /\ z e. A ) ) )
43	40, 42	ecased 1390	. . . . . . . . . . . . 13 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> ( -. x = (/) /\ z e. B ) )
44	43	simprd 114	. . . . . . . . . . . 12 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> z e. B )
45	33, 44	opelxpd 4809	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> <. x , z >. e. ( { 1o } X. B ) )
46	29, 45	sselid 3246	. . . . . . . . . 10 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> <. x , z >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
47	46, 27	eleqtrrdi 2332	. . . . . . . . 9 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x = 1o ) -> <. x , z >. e. ( A ⊔ B ) )
48	28, 47	jaodan 809	. . . . . . . 8 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ ( x = (/) \/ x = 1o ) ) -> <. x , z >. e. ( A ⊔ B ) )
49	9, 48	sylan2 286	. . . . . . 7 |- ( ( ( y = <. x , z >. /\ z e. F ) /\ x e. 2o ) -> <. x , z >. e. ( A ⊔ B ) )
50	6, 49	sylbir 135	. . . . . 6 |- ( ( y = <. x , z >. /\ ( x e. 2o /\ z e. F ) ) -> <. x , z >. e. ( A ⊔ B ) )
51	2, 50	eqeltrd 2315	. . . . 5 |- ( ( y = <. x , z >. /\ ( x e. 2o /\ z e. F ) ) -> y e. ( A ⊔ B ) )
52	51	exlimivv 1952	. . . 4 |- ( E. x E. z ( y = <. x , z >. /\ ( x e. 2o /\ z e. F ) ) -> y e. ( A ⊔ B ) )
53	1, 52	sylbi 121	. . 3 |- ( y e. U_ x e. 2o ( { x } X. F ) -> y e. ( A ⊔ B ) )
54	27	eleq2i 2305	. . . . 5 |- ( y e. ( A ⊔ B ) <-> y e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
55		elun 3370	. . . . 5 |- ( y e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( y e. ( { (/) } X. A ) \/ y e. ( { 1o } X. B ) ) )
56	54, 55	bitri 184	. . . 4 |- ( y e. ( A ⊔ B ) <-> ( y e. ( { (/) } X. A ) \/ y e. ( { 1o } X. B ) ) )
57		1st2nd2 6413	. . . . . 6 |- ( y e. ( { (/) } X. A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
58		xp1st 6403	. . . . . . . . 9 |- ( y e. ( { (/) } X. A ) -> ( 1st ` y ) e. { (/) } )
59		elsni 3727	. . . . . . . . 9 |- ( ( 1st ` y ) e. { (/) } -> ( 1st ` y ) = (/) )
60	58, 59	syl 14	. . . . . . . 8 |- ( y e. ( { (/) } X. A ) -> ( 1st ` y ) = (/) )
61		0lt2o 6718	. . . . . . . 8 |- (/) e. 2o
62	60, 61	eqeltrdi 2329	. . . . . . 7 |- ( y e. ( { (/) } X. A ) -> ( 1st ` y ) e. 2o )
63		xp2nd 6404	. . . . . . . 8 |- ( y e. ( { (/) } X. A ) -> ( 2nd ` y ) e. A )
64	60	iftrued 3647	. . . . . . . 8 |- ( y e. ( { (/) } X. A ) -> if ( ( 1st ` y ) = (/) , A , B ) = A )
65	63, 64	eleqtrrd 2318	. . . . . . 7 |- ( y e. ( { (/) } X. A ) -> ( 2nd ` y ) e. if ( ( 1st ` y ) = (/) , A , B ) )
66		eqeq1 2245	. . . . . . . . . 10 |- ( x = ( 1st ` y ) -> ( x = (/) <-> ( 1st ` y ) = (/) ) )
67	66	ifbid 3662	. . . . . . . . 9 |- ( x = ( 1st ` y ) -> if ( x = (/) , A , B ) = if ( ( 1st ` y ) = (/) , A , B ) )
68	17, 67	eqtrid 2283	. . . . . . . 8 |- ( x = ( 1st ` y ) -> F = if ( ( 1st ` y ) = (/) , A , B ) )
69	68	opeliunxp2 4922	. . . . . . 7 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. U_ x e. 2o ( { x } X. F ) <-> ( ( 1st ` y ) e. 2o /\ ( 2nd ` y ) e. if ( ( 1st ` y ) = (/) , A , B ) ) )
70	62, 65, 69	sylanbrc 421	. . . . . 6 |- ( y e. ( { (/) } X. A ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. U_ x e. 2o ( { x } X. F ) )
71	57, 70	eqeltrd 2315	. . . . 5 |- ( y e. ( { (/) } X. A ) -> y e. U_ x e. 2o ( { x } X. F ) )
72		1st2nd2 6413	. . . . . 6 |- ( y e. ( { 1o } X. B ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
73		xp1st 6403	. . . . . . . . 9 |- ( y e. ( { 1o } X. B ) -> ( 1st ` y ) e. { 1o } )
74		elsni 3727	. . . . . . . . 9 |- ( ( 1st ` y ) e. { 1o } -> ( 1st ` y ) = 1o )
75	73, 74	syl 14	. . . . . . . 8 |- ( y e. ( { 1o } X. B ) -> ( 1st ` y ) = 1o )
76		1lt2o 6719	. . . . . . . 8 |- 1o e. 2o
77	75, 76	eqeltrdi 2329	. . . . . . 7 |- ( y e. ( { 1o } X. B ) -> ( 1st ` y ) e. 2o )
78		xp2nd 6404	. . . . . . . 8 |- ( y e. ( { 1o } X. B ) -> ( 2nd ` y ) e. B )
79		eqtr2 2257	. . . . . . . . . . . 12 |- ( ( ( 1st ` y ) = 1o /\ ( 1st ` y ) = (/) ) -> 1o = (/) )
80	79	ex 115	. . . . . . . . . . 11 |- ( ( 1st ` y ) = 1o -> ( ( 1st ` y ) = (/) -> 1o = (/) ) )
81	35, 80	mtoi 674	. . . . . . . . . 10 |- ( ( 1st ` y ) = 1o -> -. ( 1st ` y ) = (/) )
82	75, 81	syl 14	. . . . . . . . 9 |- ( y e. ( { 1o } X. B ) -> -. ( 1st ` y ) = (/) )
83	82	iffalsed 3650	. . . . . . . 8 |- ( y e. ( { 1o } X. B ) -> if ( ( 1st ` y ) = (/) , A , B ) = B )
84	78, 83	eleqtrrd 2318	. . . . . . 7 |- ( y e. ( { 1o } X. B ) -> ( 2nd ` y ) e. if ( ( 1st ` y ) = (/) , A , B ) )
85	77, 84, 69	sylanbrc 421	. . . . . 6 |- ( y e. ( { 1o } X. B ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. U_ x e. 2o ( { x } X. F ) )
86	72, 85	eqeltrd 2315	. . . . 5 |- ( y e. ( { 1o } X. B ) -> y e. U_ x e. 2o ( { x } X. F ) )
87	71, 86	jaoi 728	. . . 4 |- ( ( y e. ( { (/) } X. A ) \/ y e. ( { 1o } X. B ) ) -> y e. U_ x e. 2o ( { x } X. F ) )
88	56, 87	sylbi 121	. . 3 |- ( y e. ( A ⊔ B ) -> y e. U_ x e. 2o ( { x } X. F ) )
89	53, 88	impbii 126	. 2 |- ( y e. U_ x e. 2o ( { x } X. F ) <-> y e. ( A ⊔ B ) )
90	89	eqriv 2235	1 |- U_ x e. 2o ( { x } X. F ) = ( A ⊔ B )

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   u. cun 3218   (/)c0 3520   ifcif 3638   {csn 3709   {cpr 3710   <.cop 3712   U_ciun 4012   X. cxp 4774   ` cfv 5380   1stc1st 6376   2ndc2nd 6377   1oc1o 6684   2oc2o 6685   ⊔ cdju 7383

This proof depends on 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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof 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-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-2o 6692  df-dju 7384

This theorem is used by: (None)