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Theorem pm54.43 6719
Description: Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
Assertion
Ref Expression
pm54.43  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  <->  ( A  u.  B )  ~~  2o ) )

Proof of Theorem pm54.43
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 6118 . . . . . . . 8  |-  1o  e.  On
21elexi 2622 . . . . . . 7  |-  1o  e.  _V
32ensn1 6441 . . . . . 6  |-  { 1o }  ~~  1o
43ensymi 6427 . . . . 5  |-  1o  ~~  { 1o }
5 entr 6429 . . . . 5  |-  ( ( B  ~~  1o  /\  1o  ~~  { 1o }
)  ->  B  ~~  { 1o } )
64, 5mpan2 416 . . . 4  |-  ( B 
~~  1o  ->  B  ~~  { 1o } )
71onirri 4321 . . . . . . 7  |-  -.  1o  e.  1o
8 disjsn 3478 . . . . . . 7  |-  ( ( 1o  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  1o )
97, 8mpbir 144 . . . . . 6  |-  ( 1o 
i^i  { 1o } )  =  (/)
10 unen 6461 . . . . . 6  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  (
( A  i^i  B
)  =  (/)  /\  ( 1o  i^i  { 1o }
)  =  (/) ) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
119, 10mpanr2 429 . . . . 5  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) )
1211ex 113 . . . 4  |-  ( ( A  ~~  1o  /\  B  ~~  { 1o }
)  ->  ( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) ) )
136, 12sylan2 280 . . 3  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) ) )
14 df-2o 6112 . . . . 5  |-  2o  =  suc  1o
15 df-suc 4161 . . . . 5  |-  suc  1o  =  ( 1o  u.  { 1o } )
1614, 15eqtri 2103 . . . 4  |-  2o  =  ( 1o  u.  { 1o } )
1716breq2i 3819 . . 3  |-  ( ( A  u.  B ) 
~~  2o  <->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
1813, 17syl6ibr 160 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  2o ) )
19 en1 6444 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
20 en1 6444 . . 3  |-  ( B 
~~  1o  <->  E. y  B  =  { y } )
21 1nen2 6505 . . . . . . . . . . . . 13  |-  -.  1o  ~~  2o
2221a1i 9 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -.  1o  ~~  2o )
23 unidm 3127 . . . . . . . . . . . . . . . 16  |-  ( { x }  u.  {
x } )  =  { x }
24 sneq 3433 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  { x }  =  { y } )
2524uneq2d 3138 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( { x }  u.  { x } )  =  ( { x }  u.  { y } ) )
2623, 25syl5reqr 2130 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( { x }  u.  { y } )  =  { x } )
27 vex 2615 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
2827ensn1 6441 . . . . . . . . . . . . . . 15  |-  { x }  ~~  1o
2926, 28syl6eqbr 3848 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( { x }  u.  { y } )  ~~  1o )
3029ensymd 6428 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  1o  ~~  ( { x }  u.  { y } ) )
31 entr 6429 . . . . . . . . . . . . 13  |-  ( ( 1o  ~~  ( { x }  u.  {
y } )  /\  ( { x }  u.  { y } )  ~~  2o )  ->  1o  ~~  2o )
3230, 31sylan 277 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  ( { x }  u.  { y } )  ~~  2o )  ->  1o  ~~  2o )
3322, 32mtand 624 . . . . . . . . . . 11  |-  ( x  =  y  ->  -.  ( { x }  u.  { y } )  ~~  2o )
3433necon2ai 2303 . . . . . . . . . 10  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  x  =/=  y
)
35 disjsn2 3479 . . . . . . . . . 10  |-  ( x  =/=  y  ->  ( { x }  i^i  { y } )  =  (/) )
3634, 35syl 14 . . . . . . . . 9  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) )
3736a1i 9 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) ) )
38 uneq12 3133 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  u.  B
)  =  ( { x }  u.  {
y } ) )
3938breq1d 3821 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  <->  ( { x }  u.  { y } )  ~~  2o ) )
40 ineq12 3180 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  i^i  B
)  =  ( { x }  i^i  {
y } ) )
4140eqeq1d 2091 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  i^i  B )  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
4237, 39, 413imtr4d 201 . . . . . . 7  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4342ex 113 . . . . . 6  |-  ( A  =  { x }  ->  ( B  =  {
y }  ->  (
( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4443exlimdv 1742 . . . . 5  |-  ( A  =  { x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4544exlimiv 1530 . . . 4  |-  ( E. x  A  =  {
x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4645imp 122 . . 3  |-  ( ( E. x  A  =  { x }  /\  E. y  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4719, 20, 46syl2anb 285 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4818, 47impbid 127 1  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  <->  ( A  u.  B )  ~~  2o ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434    =/= wne 2249    u. cun 2982    i^i cin 2983   (/)c0 3269   {csn 3422   class class class wbr 3811   Oncon0 4153   suc csuc 4155   1oc1o 6104   2oc2o 6105    ~~ cen 6383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-1o 6111  df-2o 6112  df-er 6220  df-en 6386
This theorem is referenced by:  pr2nelem  6720  dju1p1e2  6724
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