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Theorem djuexb 6988
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 6987 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A B )  e.  _V )
2 df-dju 6982 . . . . 5  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
32eleq1i 2223 . . . 4  |-  ( ( A B )  e.  _V  <->  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )  e.  _V )
4 unexb 4402 . . . 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 4091 . . . . . . 7  |-  (/)  e.  _V
76snm 3679 . . . . . 6  |-  E. x  x  e.  { (/) }
8 rnxpm 5015 . . . . . 6  |-  ( E. x  x  e.  { (/)
}  ->  ran  ( {
(/) }  X.  A
)  =  A )
97, 8ax-mp 5 . . . . 5  |-  ran  ( { (/) }  X.  A
)  =  A
10 rnexg 4851 . . . . 5  |-  ( ( { (/) }  X.  A
)  e.  _V  ->  ran  ( { (/) }  X.  A )  e.  _V )
119, 10eqeltrrid 2245 . . . 4  |-  ( ( { (/) }  X.  A
)  e.  _V  ->  A  e.  _V )
12 1oex 6371 . . . . . . 7  |-  1o  e.  _V
1312snm 3679 . . . . . 6  |-  E. x  x  e.  { 1o }
14 rnxpm 5015 . . . . . 6  |-  ( E. x  x  e.  { 1o }  ->  ran  ( { 1o }  X.  B
)  =  B )
1513, 14ax-mp 5 . . . . 5  |-  ran  ( { 1o }  X.  B
)  =  B
16 rnexg 4851 . . . . 5  |-  ( ( { 1o }  X.  B )  e.  _V  ->  ran  ( { 1o }  X.  B )  e. 
_V )
1715, 16eqeltrrid 2245 . . . 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 1335   E.wex 1472    e. wcel 2128   _Vcvv 2712    u. cun 3100   (/)c0 3394   {csn 3560    X. cxp 4584   ran crn 4587   1oc1o 6356   ⊔ cdju 6981
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-1o 6363  df-dju 6982
This theorem is referenced by:  ctfoex  7062
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