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Theorem djucomen 7065
Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
djucomen  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  ~~  ( B A )
)

Proof of Theorem djucomen
StepHypRef Expression
1 1oex 6314 . . . 4  |-  1o  e.  _V
2 xpsnen2g 6716 . . . 4  |-  ( ( 1o  e.  _V  /\  A  e.  V )  ->  ( { 1o }  X.  A )  ~~  A
)
31, 2mpan 420 . . 3  |-  ( A  e.  V  ->  ( { 1o }  X.  A
)  ~~  A )
4 0ex 4050 . . . 4  |-  (/)  e.  _V
5 xpsnen2g 6716 . . . 4  |-  ( (
(/)  e.  _V  /\  B  e.  W )  ->  ( { (/) }  X.  B
)  ~~  B )
64, 5mpan 420 . . 3  |-  ( B  e.  W  ->  ( { (/) }  X.  B
)  ~~  B )
7 ensym 6668 . . . 4  |-  ( ( { 1o }  X.  A )  ~~  A  ->  A  ~~  ( { 1o }  X.  A
) )
8 ensym 6668 . . . 4  |-  ( ( { (/) }  X.  B
)  ~~  B  ->  B 
~~  ( { (/) }  X.  B ) )
9 incom 3263 . . . . . 6  |-  ( ( { 1o }  X.  A )  i^i  ( { (/) }  X.  B
) )  =  ( ( { (/) }  X.  B )  i^i  ( { 1o }  X.  A
) )
10 xp01disjl 6324 . . . . . 6  |-  ( ( { (/) }  X.  B
)  i^i  ( { 1o }  X.  A ) )  =  (/)
119, 10eqtri 2158 . . . . 5  |-  ( ( { 1o }  X.  A )  i^i  ( { (/) }  X.  B
) )  =  (/)
12 djuenun 7061 . . . . 5  |-  ( ( A  ~~  ( { 1o }  X.  A
)  /\  B  ~~  ( { (/) }  X.  B
)  /\  ( ( { 1o }  X.  A
)  i^i  ( { (/)
}  X.  B ) )  =  (/) )  -> 
( A B )  ~~  ( ( { 1o }  X.  A )  u.  ( { (/) }  X.  B ) ) )
1311, 12mp3an3 1304 . . . 4  |-  ( ( A  ~~  ( { 1o }  X.  A
)  /\  B  ~~  ( { (/) }  X.  B
) )  ->  ( A B )  ~~  (
( { 1o }  X.  A )  u.  ( { (/) }  X.  B
) ) )
147, 8, 13syl2an 287 . . 3  |-  ( ( ( { 1o }  X.  A )  ~~  A  /\  ( { (/) }  X.  B )  ~~  B
)  ->  ( A B )  ~~  ( ( { 1o }  X.  A )  u.  ( { (/) }  X.  B
) ) )
153, 6, 14syl2an 287 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  ~~  ( ( { 1o }  X.  A )  u.  ( { (/) }  X.  B ) ) )
16 df-dju 6916 . . 3  |-  ( B A )  =  ( ( { (/) }  X.  B )  u.  ( { 1o }  X.  A
) )
1716equncomi 3217 . 2  |-  ( B A )  =  ( ( { 1o }  X.  A )  u.  ( { (/) }  X.  B
) )
1815, 17breqtrrdi 3965 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  ~~  ( B A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2681    u. cun 3064    i^i cin 3065   (/)c0 3358   {csn 3522   class class class wbr 3924    X. cxp 4532   1oc1o 6299    ~~ cen 6625   ⊔ cdju 6915
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-er 6422  df-en 6628  df-dju 6916  df-inl 6925  df-inr 6926
This theorem is referenced by: (None)
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