Theorem djuexb 6929

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 6928	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A ⊔ B ) e. _V )
2		df-dju 6923	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
3	2	eleq1i 2205	. . . 4 |- ( ( A ⊔ B ) e. _V <-> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) e. _V )
4		unexb 4363	. . . 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 4055	. . . . . . 7 |- (/) e. _V
7	6	snm 3643	. . . . . 6 |- E. x x e. { (/) }
8		rnxpm 4968	. . . . . 6 |- ( E. x x e. { (/) } -> ran ( { (/) } X. A ) = A )
9	7, 8	ax-mp 5	. . . . 5 |- ran ( { (/) } X. A ) = A
10		rnexg 4804	. . . . 5 |- ( ( { (/) } X. A ) e. _V -> ran ( { (/) } X. A ) e. _V )
11	9, 10	eqeltrrid 2227	. . . 4 |- ( ( { (/) } X. A ) e. _V -> A e. _V )
12		1oex 6321	. . . . . . 7 |- 1o e. _V
13	12	snm 3643	. . . . . 6 |- E. x x e. { 1o }
14		rnxpm 4968	. . . . . 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 4804	. . . . 5 |- ( ( { 1o } X. B ) e. _V -> ran ( { 1o } X. B ) e. _V )
17	15, 16	eqeltrrid 2227	. . . 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 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686   u. cun 3069   (/)c0 3363   {csn 3527   X. cxp 4537   ran crn 4540   1oc1o 6306   ⊔ cdju 6922

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-1o 6313  df-dju 6923

This theorem is referenced by:  ctfoex  7003