Theorem djuexb 7374

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 7373	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A ⊔ B ) e. _V )
2		df-dju 7368	. . . . 5 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
3	2	eleq1i 2304	. . . 4 |- ( ( A ⊔ B ) e. _V <-> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) e. _V )
4		unexb 4583	. . . 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 4255	. . . . . . 7 |- (/) e. _V
7	6	snm 3828	. . . . . 6 |- E. x x e. { (/) }
8		rnxpm 5212	. . . . . 6 |- ( E. x x e. { (/) } -> ran ( { (/) } X. A ) = A )
9	7, 8	ax-mp 5	. . . . 5 |- ran ( { (/) } X. A ) = A
10		rnexg 5042	. . . . 5 |- ( ( { (/) } X. A ) e. _V -> ran ( { (/) } X. A ) e. _V )
11	9, 10	eqeltrrid 2326	. . . 4 |- ( ( { (/) } X. A ) e. _V -> A e. _V )
12		1oex 6685	. . . . . . 7 |- 1o e. _V
13	12	snm 3828	. . . . . 6 |- E. x x e. { 1o }
14		rnxpm 5212	. . . . . 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 5042	. . . . 5 |- ( ( { 1o } X. B ) e. _V -> ran ( { 1o } X. B ) e. _V )
17	15, 16	eqeltrrid 2326	. . . 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   u. cun 3218   (/)c0 3520   {csn 3705   X. cxp 4767   ran crn 4770   1oc1o 6670   ⊔ cdju 7367

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem 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-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-1o 6677  df-dju 7368

This theorem is referenced by:  ctfoex  7448