Theorem djuexb 6988

Description: The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)

Assertion

Ref	Expression
djuexb	|- ( ( A e. _V /\ B e. _V ) <-> ( A ⊔ B ) e. _V )

Proof of Theorem djuexb

Step	Hyp	Ref	Expression
1		djuex 6987	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A ⊔ B ) e. _V )
2		df-dju 6982	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
3	2	eleq1i 2223	. . . 4 |- ( ( A ⊔ B ) e. _V <-> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) e. _V )
4		unexb 4402	. . . 4 |- ( ( ( { (/) } X. A ) e. _V /\ ( { 1o } X. B ) e. _V ) <-> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) e. _V )
5	3, 4	bitr4i 186	. . 3 |- ( ( A ⊔ B ) e. _V <-> ( ( { (/) } X. A ) e. _V /\ ( { 1o } X. B ) e. _V ) )
6		0ex 4091	. . . . . . 7 |- (/) e. _V
7	6	snm 3679	. . . . . 6 |- E. x x e. { (/) }
8		rnxpm 5015	. . . . . 6 |- ( E. x x e. { (/) } -> ran ( { (/) } X. A ) = A )
9	7, 8	ax-mp 5	. . . . 5 |- ran ( { (/) } X. A ) = A
10		rnexg 4851	. . . . 5 |- ( ( { (/) } X. A ) e. _V -> ran ( { (/) } X. A ) e. _V )
11	9, 10	eqeltrrid 2245	. . . 4 |- ( ( { (/) } X. A ) e. _V -> A e. _V )
12		1oex 6371	. . . . . . 7 |- 1o e. _V
13	12	snm 3679	. . . . . 6 |- E. x x e. { 1o }
14		rnxpm 5015	. . . . . 6 |- ( E. x x e. { 1o } -> ran ( { 1o } X. B ) = B )
15	13, 14	ax-mp 5	. . . . 5 |- ran ( { 1o } X. B ) = B
16		rnexg 4851	. . . . 5 |- ( ( { 1o } X. B ) e. _V -> ran ( { 1o } X. B ) e. _V )
17	15, 16	eqeltrrid 2245	. . . 4 |- ( ( { 1o } X. B ) e. _V -> B e. _V )
18	11, 17	anim12i 336	. . 3 |- ( ( ( { (/) } X. A ) e. _V /\ ( { 1o } X. B ) e. _V ) -> ( A e. _V /\ B e. _V ) )
19	5, 18	sylbi 120	. 2 |- ( ( A ⊔ B ) e. _V -> ( A e. _V /\ B e. _V ) )
20	1, 19	impbii 125	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A ⊔ B ) e. _V )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128   _Vcvv 2712   u. cun 3100   (/)c0 3394   {csn 3560   X. cxp 4584   ran crn 4587   1oc1o 6356   ⊔ cdju 6981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-1o 6363  df-dju 6982

This theorem is referenced by:  ctfoex  7062