Theorem djuexb 7148

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 7147	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A ⊔ B ) e. _V )
2		df-dju 7142	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
3	2	eleq1i 2271	. . . 4 |- ( ( A ⊔ B ) e. _V <-> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) e. _V )
4		unexb 4490	. . . 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 4172	. . . . . . 7 |- (/) e. _V
7	6	snm 3753	. . . . . 6 |- E. x x e. { (/) }
8		rnxpm 5113	. . . . . 6 |- ( E. x x e. { (/) } -> ran ( { (/) } X. A ) = A )
9	7, 8	ax-mp 5	. . . . 5 |- ran ( { (/) } X. A ) = A
10		rnexg 4944	. . . . 5 |- ( ( { (/) } X. A ) e. _V -> ran ( { (/) } X. A ) e. _V )
11	9, 10	eqeltrrid 2293	. . . 4 |- ( ( { (/) } X. A ) e. _V -> A e. _V )
12		1oex 6512	. . . . . . 7 |- 1o e. _V
13	12	snm 3753	. . . . . 6 |- E. x x e. { 1o }
14		rnxpm 5113	. . . . . 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 4944	. . . . 5 |- ( ( { 1o } X. B ) e. _V -> ran ( { 1o } X. B ) e. _V )
17	15, 16	eqeltrrid 2293	. . . 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 1373   E.wex 1515   e. wcel 2176   _Vcvv 2772   u. cun 3164   (/)c0 3460   {csn 3633   X. cxp 4674   ran crn 4677   1oc1o 6497   ⊔ cdju 7141

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-tr 4144  df-iord 4414  df-on 4416  df-suc 4419  df-xp 4682  df-rel 4683  df-cnv 4684  df-dm 4686  df-rn 4687  df-1o 6504  df-dju 7142

This theorem is referenced by:  ctfoex  7222