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Theorem djuassen 7037
Description: Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
djuassen  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A B ) C )  ~~  ( A ( B C )
) )

Proof of Theorem djuassen
StepHypRef Expression
1 0ex 4023 . . . . . 6  |-  (/)  e.  _V
2 simp1 964 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  e.  V )
3 xpsnen2g 6689 . . . . . 6  |-  ( (
(/)  e.  _V  /\  A  e.  V )  ->  ( { (/) }  X.  A
)  ~~  A )
41, 2, 3sylancr 408 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { (/) }  X.  A )  ~~  A
)
54ensymd 6643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  ( {
(/) }  X.  A
) )
6 1oex 6287 . . . . . . 7  |-  1o  e.  _V
71snex 4077 . . . . . . . 8  |-  { (/) }  e.  _V
8 simp2 965 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
9 xpexg 4621 . . . . . . . 8  |-  ( ( { (/) }  e.  _V  /\  B  e.  W )  ->  ( { (/) }  X.  B )  e. 
_V )
107, 8, 9sylancr 408 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { (/) }  X.  B )  e.  _V )
11 xpsnen2g 6689 . . . . . . 7  |-  ( ( 1o  e.  _V  /\  ( { (/) }  X.  B
)  e.  _V )  ->  ( { 1o }  X.  ( { (/) }  X.  B ) )  ~~  ( { (/) }  X.  B
) )
126, 10, 11sylancr 408 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { 1o }  X.  ( { (/) }  X.  B ) )  ~~  ( { (/) }  X.  B
) )
13 xpsnen2g 6689 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  B  e.  W )  ->  ( { (/) }  X.  B
)  ~~  B )
141, 8, 13sylancr 408 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { (/) }  X.  B )  ~~  B
)
15 entr 6644 . . . . . 6  |-  ( ( ( { 1o }  X.  ( { (/) }  X.  B ) )  ~~  ( { (/) }  X.  B
)  /\  ( { (/)
}  X.  B ) 
~~  B )  -> 
( { 1o }  X.  ( { (/) }  X.  B ) )  ~~  B )
1612, 14, 15syl2anc 406 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { 1o }  X.  ( { (/) }  X.  B ) )  ~~  B )
1716ensymd 6643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( { 1o }  X.  ( { (/) }  X.  B
) ) )
18 xp01disjl 6297 . . . . 5  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  ( {
(/) }  X.  B
) ) )  =  (/)
1918a1i 9 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( { (/) }  X.  A )  i^i  ( { 1o }  X.  ( { (/) }  X.  B ) ) )  =  (/) )
20 djuenun 7032 . . . 4  |-  ( ( A  ~~  ( {
(/) }  X.  A
)  /\  B  ~~  ( { 1o }  X.  ( { (/) }  X.  B
) )  /\  (
( { (/) }  X.  A )  i^i  ( { 1o }  X.  ( { (/) }  X.  B
) ) )  =  (/) )  ->  ( A B )  ~~  (
( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B
) ) ) )
215, 17, 19, 20syl3anc 1199 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A B )  ~~  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B ) ) ) )
226snex 4077 . . . . . . 7  |-  { 1o }  e.  _V
23 simp3 966 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
24 xpexg 4621 . . . . . . 7  |-  ( ( { 1o }  e.  _V  /\  C  e.  X
)  ->  ( { 1o }  X.  C )  e.  _V )
2522, 23, 24sylancr 408 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { 1o }  X.  C )  e.  _V )
26 xpsnen2g 6689 . . . . . 6  |-  ( ( 1o  e.  _V  /\  ( { 1o }  X.  C )  e.  _V )  ->  ( { 1o }  X.  ( { 1o }  X.  C ) ) 
~~  ( { 1o }  X.  C ) )
276, 25, 26sylancr 408 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { 1o }  X.  ( { 1o }  X.  C ) )  ~~  ( { 1o }  X.  C ) )
28 xpsnen2g 6689 . . . . . 6  |-  ( ( 1o  e.  _V  /\  C  e.  X )  ->  ( { 1o }  X.  C )  ~~  C
)
296, 23, 28sylancr 408 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { 1o }  X.  C )  ~~  C
)
30 entr 6644 . . . . 5  |-  ( ( ( { 1o }  X.  ( { 1o }  X.  C ) )  ~~  ( { 1o }  X.  C )  /\  ( { 1o }  X.  C
)  ~~  C )  ->  ( { 1o }  X.  ( { 1o }  X.  C ) )  ~~  C )
3127, 29, 30syl2anc 406 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( { 1o }  X.  ( { 1o }  X.  C ) )  ~~  C )
3231ensymd 6643 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( { 1o }  X.  ( { 1o }  X.  C
) ) )
33 indir 3293 . . . . 5  |-  ( ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B
) ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) )  =  ( ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  ( { 1o }  X.  C
) ) )  u.  ( ( { 1o }  X.  ( { (/) }  X.  B ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) ) )
34 xp01disjl 6297 . . . . . . 7  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  ( { 1o }  X.  C
) ) )  =  (/)
35 xp01disjl 6297 . . . . . . . . 9  |-  ( ( { (/) }  X.  B
)  i^i  ( { 1o }  X.  C ) )  =  (/)
3635xpeq2i 4528 . . . . . . . 8  |-  ( { 1o }  X.  (
( { (/) }  X.  B )  i^i  ( { 1o }  X.  C
) ) )  =  ( { 1o }  X.  (/) )
37 xpindi 4642 . . . . . . . 8  |-  ( { 1o }  X.  (
( { (/) }  X.  B )  i^i  ( { 1o }  X.  C
) ) )  =  ( ( { 1o }  X.  ( { (/) }  X.  B ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) )
38 xp0 4926 . . . . . . . 8  |-  ( { 1o }  X.  (/) )  =  (/)
3936, 37, 383eqtr3i 2144 . . . . . . 7  |-  ( ( { 1o }  X.  ( { (/) }  X.  B
) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C
) ) )  =  (/)
4034, 39uneq12i 3196 . . . . . 6  |-  ( ( ( { (/) }  X.  A )  i^i  ( { 1o }  X.  ( { 1o }  X.  C
) ) )  u.  ( ( { 1o }  X.  ( { (/) }  X.  B ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) ) )  =  (
(/)  u.  (/) )
41 un0 3364 . . . . . 6  |-  ( (/)  u.  (/) )  =  (/)
4240, 41eqtri 2136 . . . . 5  |-  ( ( ( { (/) }  X.  A )  i^i  ( { 1o }  X.  ( { 1o }  X.  C
) ) )  u.  ( ( { 1o }  X.  ( { (/) }  X.  B ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) ) )  =  (/)
4333, 42eqtri 2136 . . . 4  |-  ( ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B
) ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) )  =  (/)
4443a1i 9 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  ( { (/) }  X.  B
) ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) )  =  (/) )
45 djuenun 7032 . . 3  |-  ( ( ( A B )  ~~  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B ) ) )  /\  C  ~~  ( { 1o }  X.  ( { 1o }  X.  C
) )  /\  (
( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B ) ) )  i^i  ( { 1o }  X.  ( { 1o }  X.  C ) ) )  =  (/) )  -> 
( ( A B ) C )  ~~  (
( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B ) ) )  u.  ( { 1o }  X.  ( { 1o }  X.  C ) ) ) )
4621, 32, 44, 45syl3anc 1199 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A B ) C )  ~~  (
( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B ) ) )  u.  ( { 1o }  X.  ( { 1o }  X.  C ) ) ) )
47 df-dju 6889 . . . . . 6  |-  ( B C )  =  ( ( { (/) }  X.  B )  u.  ( { 1o }  X.  C
) )
4847xpeq2i 4528 . . . . 5  |-  ( { 1o }  X.  ( B C ) )  =  ( { 1o }  X.  ( ( { (/) }  X.  B )  u.  ( { 1o }  X.  C ) ) )
49 xpundi 4563 . . . . 5  |-  ( { 1o }  X.  (
( { (/) }  X.  B )  u.  ( { 1o }  X.  C
) ) )  =  ( ( { 1o }  X.  ( { (/) }  X.  B ) )  u.  ( { 1o }  X.  ( { 1o }  X.  C ) ) )
5048, 49eqtri 2136 . . . 4  |-  ( { 1o }  X.  ( B C ) )  =  ( ( { 1o }  X.  ( { (/) }  X.  B ) )  u.  ( { 1o }  X.  ( { 1o }  X.  C ) ) )
5150uneq2i 3195 . . 3  |-  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  ( B C ) ) )  =  ( ( {
(/) }  X.  A
)  u.  ( ( { 1o }  X.  ( { (/) }  X.  B
) )  u.  ( { 1o }  X.  ( { 1o }  X.  C
) ) ) )
52 df-dju 6889 . . 3  |-  ( A ( B C ) )  =  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  ( B C ) ) )
53 unass 3201 . . 3  |-  ( ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  ( { (/) }  X.  B
) ) )  u.  ( { 1o }  X.  ( { 1o }  X.  C ) ) )  =  ( ( {
(/) }  X.  A
)  u.  ( ( { 1o }  X.  ( { (/) }  X.  B
) )  u.  ( { 1o }  X.  ( { 1o }  X.  C
) ) ) )
5451, 52, 533eqtr4i 2146 . 2  |-  ( A ( B C ) )  =  ( ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  ( { (/) }  X.  B
) ) )  u.  ( { 1o }  X.  ( { 1o }  X.  C ) ) )
5546, 54breqtrrdi 3938 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A B ) C )  ~~  ( A ( B C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 945    = wceq 1314    e. wcel 1463   _Vcvv 2658    u. cun 3037    i^i cin 3038   (/)c0 3331   {csn 3495   class class class wbr 3897    X. cxp 4505   1oc1o 6272    ~~ cen 6598   ⊔ cdju 6888
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-er 6395  df-en 6601  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by: (None)
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