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Theorem djuexb 7021
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
StepHypRef Expression
1 djuex 7020 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A B )  e.  _V )
2 df-dju 7015 . . . . 5  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
32eleq1i 2236 . . . 4  |-  ( ( A B )  e.  _V  <->  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )  e.  _V )
4 unexb 4427 . . . 4  |-  ( ( ( { (/) }  X.  A )  e.  _V  /\  ( { 1o }  X.  B )  e.  _V ) 
<->  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) )  e. 
_V )
53, 4bitr4i 186 . . 3  |-  ( ( A B )  e.  _V  <->  ( ( { (/) }  X.  A )  e.  _V  /\  ( { 1o }  X.  B )  e.  _V ) )
6 0ex 4116 . . . . . . 7  |-  (/)  e.  _V
76snm 3703 . . . . . 6  |-  E. x  x  e.  { (/) }
8 rnxpm 5040 . . . . . 6  |-  ( E. x  x  e.  { (/)
}  ->  ran  ( {
(/) }  X.  A
)  =  A )
97, 8ax-mp 5 . . . . 5  |-  ran  ( { (/) }  X.  A
)  =  A
10 rnexg 4876 . . . . 5  |-  ( ( { (/) }  X.  A
)  e.  _V  ->  ran  ( { (/) }  X.  A )  e.  _V )
119, 10eqeltrrid 2258 . . . 4  |-  ( ( { (/) }  X.  A
)  e.  _V  ->  A  e.  _V )
12 1oex 6403 . . . . . . 7  |-  1o  e.  _V
1312snm 3703 . . . . . 6  |-  E. x  x  e.  { 1o }
14 rnxpm 5040 . . . . . 6  |-  ( E. x  x  e.  { 1o }  ->  ran  ( { 1o }  X.  B
)  =  B )
1513, 14ax-mp 5 . . . . 5  |-  ran  ( { 1o }  X.  B
)  =  B
16 rnexg 4876 . . . . 5  |-  ( ( { 1o }  X.  B )  e.  _V  ->  ran  ( { 1o }  X.  B )  e. 
_V )
1715, 16eqeltrrid 2258 . . . 4  |-  ( ( { 1o }  X.  B )  e.  _V  ->  B  e.  _V )
1811, 17anim12i 336 . . 3  |-  ( ( ( { (/) }  X.  A )  e.  _V  /\  ( { 1o }  X.  B )  e.  _V )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
195, 18sylbi 120 . 2  |-  ( ( A B )  e.  _V  ->  ( A  e.  _V  /\  B  e.  _V )
)
201, 19impbii 125 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730    u. cun 3119   (/)c0 3414   {csn 3583    X. cxp 4609   ran crn 4612   1oc1o 6388   ⊔ cdju 7014
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-1o 6395  df-dju 7015
This theorem is referenced by:  ctfoex  7095
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