Theorem djuexb 7211

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 7210	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A ⊔ B ) e. _V )
2		df-dju 7205	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
3	2	eleq1i 2295	. . . 4 |- ( ( A ⊔ B ) e. _V <-> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) e. _V )
4		unexb 4533	. . . 4 |- ( ( ( { (/) } X. A ) e. _V /\ ( { 1o } X. B ) e. _V ) <-> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) e. _V )
5	3, 4	bitr4i 187	. . 3 |- ( ( A ⊔ B ) e. _V <-> ( ( { (/) } X. A ) e. _V /\ ( { 1o } X. B ) e. _V ) )
6		0ex 4211	. . . . . . 7 |- (/) e. _V
7	6	snm 3787	. . . . . 6 |- E. x x e. { (/) }
8		rnxpm 5158	. . . . . 6 |- ( E. x x e. { (/) } -> ran ( { (/) } X. A ) = A )
9	7, 8	ax-mp 5	. . . . 5 |- ran ( { (/) } X. A ) = A
10		rnexg 4989	. . . . 5 |- ( ( { (/) } X. A ) e. _V -> ran ( { (/) } X. A ) e. _V )
11	9, 10	eqeltrrid 2317	. . . 4 |- ( ( { (/) } X. A ) e. _V -> A e. _V )
12		1oex 6570	. . . . . . 7 |- 1o e. _V
13	12	snm 3787	. . . . . 6 |- E. x x e. { 1o }
14		rnxpm 5158	. . . . . 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 4989	. . . . 5 |- ( ( { 1o } X. B ) e. _V -> ran ( { 1o } X. B ) e. _V )
17	15, 16	eqeltrrid 2317	. . . 4 |- ( ( { 1o } X. B ) e. _V -> B e. _V )
18	11, 17	anim12i 338	. . 3 |- ( ( ( { (/) } X. A ) e. _V /\ ( { 1o } X. B ) e. _V ) -> ( A e. _V /\ B e. _V ) )
19	5, 18	sylbi 121	. 2 |- ( ( A ⊔ B ) e. _V -> ( A e. _V /\ B e. _V ) )
20	1, 19	impbii 126	1 |- ( ( A e. _V /\ B e. _V ) <-> ( A ⊔ B ) e. _V )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   u. cun 3195   (/)c0 3491   {csn 3666   X. cxp 4717   ran crn 4720   1oc1o 6555   ⊔ cdju 7204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-1o 6562  df-dju 7205

This theorem is referenced by:  ctfoex  7285