ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm54.43 Unicode version

Theorem pm54.43 7014
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 6288 . . . . . . . 8  |-  1o  e.  On
21elexi 2672 . . . . . . 7  |-  1o  e.  _V
32ensn1 6658 . . . . . 6  |-  { 1o }  ~~  1o
43ensymi 6644 . . . . 5  |-  1o  ~~  { 1o }
5 entr 6646 . . . . 5  |-  ( ( B  ~~  1o  /\  1o  ~~  { 1o }
)  ->  B  ~~  { 1o } )
64, 5mpan2 421 . . . 4  |-  ( B 
~~  1o  ->  B  ~~  { 1o } )
71onirri 4428 . . . . . . 7  |-  -.  1o  e.  1o
8 disjsn 3555 . . . . . . 7  |-  ( ( 1o  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  1o )
97, 8mpbir 145 . . . . . 6  |-  ( 1o 
i^i  { 1o } )  =  (/)
10 unen 6678 . . . . . 6  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  (
( A  i^i  B
)  =  (/)  /\  ( 1o  i^i  { 1o }
)  =  (/) ) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
119, 10mpanr2 434 . . . . 5  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) )
1211ex 114 . . . 4  |-  ( ( A  ~~  1o  /\  B  ~~  { 1o }
)  ->  ( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) ) )
136, 12sylan2 284 . . 3  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) ) )
14 df-2o 6282 . . . . 5  |-  2o  =  suc  1o
15 df-suc 4263 . . . . 5  |-  suc  1o  =  ( 1o  u.  { 1o } )
1614, 15eqtri 2138 . . . 4  |-  2o  =  ( 1o  u.  { 1o } )
1716breq2i 3907 . . 3  |-  ( ( A  u.  B ) 
~~  2o  <->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
1813, 17syl6ibr 161 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  2o ) )
19 en1 6661 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
20 en1 6661 . . 3  |-  ( B 
~~  1o  <->  E. y  B  =  { y } )
21 1nen2 6723 . . . . . . . . . . . . 13  |-  -.  1o  ~~  2o
2221a1i 9 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -.  1o  ~~  2o )
23 unidm 3189 . . . . . . . . . . . . . . . 16  |-  ( { x }  u.  {
x } )  =  { x }
24 sneq 3508 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  { x }  =  { y } )
2524uneq2d 3200 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( { x }  u.  { x } )  =  ( { x }  u.  { y } ) )
2623, 25syl5reqr 2165 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( { x }  u.  { y } )  =  { x } )
27 vex 2663 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
2827ensn1 6658 . . . . . . . . . . . . . . 15  |-  { x }  ~~  1o
2926, 28eqbrtrdi 3937 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( { x }  u.  { y } )  ~~  1o )
3029ensymd 6645 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  1o  ~~  ( { x }  u.  { y } ) )
31 entr 6646 . . . . . . . . . . . . 13  |-  ( ( 1o  ~~  ( { x }  u.  {
y } )  /\  ( { x }  u.  { y } )  ~~  2o )  ->  1o  ~~  2o )
3230, 31sylan 281 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  ( { x }  u.  { y } )  ~~  2o )  ->  1o  ~~  2o )
3322, 32mtand 639 . . . . . . . . . . 11  |-  ( x  =  y  ->  -.  ( { x }  u.  { y } )  ~~  2o )
3433necon2ai 2339 . . . . . . . . . 10  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  x  =/=  y
)
35 disjsn2 3556 . . . . . . . . . 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 3195 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  u.  B
)  =  ( { x }  u.  {
y } ) )
3938breq1d 3909 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  <->  ( { x }  u.  { y } )  ~~  2o ) )
40 ineq12 3242 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  i^i  B
)  =  ( { x }  i^i  {
y } ) )
4140eqeq1d 2126 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  i^i  B )  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
4237, 39, 413imtr4d 202 . . . . . . 7  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4342ex 114 . . . . . 6  |-  ( A  =  { x }  ->  ( B  =  {
y }  ->  (
( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4443exlimdv 1775 . . . . 5  |-  ( A  =  { x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4544exlimiv 1562 . . . 4  |-  ( E. x  A  =  {
x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4645imp 123 . . 3  |-  ( ( E. x  A  =  { x }  /\  E. y  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4719, 20, 46syl2anb 289 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4818, 47impbid 128 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 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465    =/= wne 2285    u. cun 3039    i^i cin 3040   (/)c0 3333   {csn 3497   class class class wbr 3899   Oncon0 4255   suc csuc 4257   1oc1o 6274   2oc2o 6275    ~~ cen 6600
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1o 6281  df-2o 6282  df-er 6397  df-en 6603
This theorem is referenced by:  pr2nelem  7015  dju1p1e2  7021
  Copyright terms: Public domain W3C validator