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Theorem endjudisj 7082
Description: Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
Assertion
Ref Expression
endjudisj  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A B )  ~~  ( A  u.  B )
)

Proof of Theorem endjudisj
StepHypRef Expression
1 djuun 6959 . 2  |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )
2 eninl 6989 . . . 4  |-  ( A  e.  V  ->  (inl " A )  ~~  A
)
323ad2ant1 1003 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  (inl " A )  ~~  A
)
4 eninr 6990 . . . 4  |-  ( B  e.  W  ->  (inr " B )  ~~  B
)
543ad2ant2 1004 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  (inr " B )  ~~  B
)
6 djuin 6956 . . . 4  |-  ( (inl " A )  i^i  (inr " B ) )  =  (/)
76a1i 9 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  (
(inl " A )  i^i  (inr " B ) )  =  (/) )
8 simp3 984 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A  i^i  B )  =  (/) )
9 unen 6717 . . 3  |-  ( ( ( (inl " A
)  ~~  A  /\  (inr " B )  ~~  B )  /\  (
( (inl " A
)  i^i  (inr " B
) )  =  (/)  /\  ( A  i^i  B
)  =  (/) ) )  ->  ( (inl " A )  u.  (inr " B ) )  ~~  ( A  u.  B
) )
103, 5, 7, 8, 9syl22anc 1218 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  (
(inl " A )  u.  (inr " B ) )  ~~  ( A  u.  B ) )
111, 10eqbrtrrid 3971 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A B )  ~~  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963    = wceq 1332    e. wcel 1481    u. cun 3073    i^i cin 3074   (/)c0 3367   class class class wbr 3936   "cima 4549    ~~ cen 6639   ⊔ cdju 6929  inlcinl 6937  inrcinr 6938
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1st 6045  df-2nd 6046  df-1o 6320  df-er 6436  df-en 6642  df-dju 6930  df-inl 6939  df-inr 6940
This theorem is referenced by:  djuenun  7084  dju0en  7086  exmidunben  11973
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