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Theorem pm54.43 7012
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 6286 . . . . . . . 8  |-  1o  e.  On
21elexi 2670 . . . . . . 7  |-  1o  e.  _V
32ensn1 6656 . . . . . 6  |-  { 1o }  ~~  1o
43ensymi 6642 . . . . 5  |-  1o  ~~  { 1o }
5 entr 6644 . . . . 5  |-  ( ( B  ~~  1o  /\  1o  ~~  { 1o }
)  ->  B  ~~  { 1o } )
64, 5mpan2 419 . . . 4  |-  ( B 
~~  1o  ->  B  ~~  { 1o } )
71onirri 4426 . . . . . . 7  |-  -.  1o  e.  1o
8 disjsn 3553 . . . . . . 7  |-  ( ( 1o  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  1o )
97, 8mpbir 145 . . . . . 6  |-  ( 1o 
i^i  { 1o } )  =  (/)
10 unen 6676 . . . . . 6  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  (
( A  i^i  B
)  =  (/)  /\  ( 1o  i^i  { 1o }
)  =  (/) ) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
119, 10mpanr2 432 . . . . 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 282 . . 3  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) ) )
14 df-2o 6280 . . . . 5  |-  2o  =  suc  1o
15 df-suc 4261 . . . . 5  |-  suc  1o  =  ( 1o  u.  { 1o } )
1614, 15eqtri 2136 . . . 4  |-  2o  =  ( 1o  u.  { 1o } )
1716breq2i 3905 . . 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 6659 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
20 en1 6659 . . 3  |-  ( B 
~~  1o  <->  E. y  B  =  { y } )
21 1nen2 6721 . . . . . . . . . . . . 13  |-  -.  1o  ~~  2o
2221a1i 9 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -.  1o  ~~  2o )
23 unidm 3187 . . . . . . . . . . . . . . . 16  |-  ( { x }  u.  {
x } )  =  { x }
24 sneq 3506 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  { x }  =  { y } )
2524uneq2d 3198 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( { x }  u.  { x } )  =  ( { x }  u.  { y } ) )
2623, 25syl5reqr 2163 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( { x }  u.  { y } )  =  { x } )
27 vex 2661 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
2827ensn1 6656 . . . . . . . . . . . . . . 15  |-  { x }  ~~  1o
2926, 28eqbrtrdi 3935 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( { x }  u.  { y } )  ~~  1o )
3029ensymd 6643 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  1o  ~~  ( { x }  u.  { y } ) )
31 entr 6644 . . . . . . . . . . . . 13  |-  ( ( 1o  ~~  ( { x }  u.  {
y } )  /\  ( { x }  u.  { y } )  ~~  2o )  ->  1o  ~~  2o )
3230, 31sylan 279 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  ( { x }  u.  { y } )  ~~  2o )  ->  1o  ~~  2o )
3322, 32mtand 637 . . . . . . . . . . 11  |-  ( x  =  y  ->  -.  ( { x }  u.  { y } )  ~~  2o )
3433necon2ai 2337 . . . . . . . . . 10  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  x  =/=  y
)
35 disjsn2 3554 . . . . . . . . . 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 3193 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  u.  B
)  =  ( { x }  u.  {
y } ) )
3938breq1d 3907 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  <->  ( { x }  u.  { y } )  ~~  2o ) )
40 ineq12 3240 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  i^i  B
)  =  ( { x }  i^i  {
y } ) )
4140eqeq1d 2124 . . . . . . . 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 1773 . . . . 5  |-  ( A  =  { x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4544exlimiv 1560 . . . 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 287 . 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 1314   E.wex 1451    e. wcel 1463    =/= wne 2283    u. cun 3037    i^i cin 3038   (/)c0 3331   {csn 3495   class class class wbr 3897   Oncon0 4253   suc csuc 4255   1oc1o 6272   2oc2o 6273    ~~ cen 6598
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-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  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-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-br 3898  df-opab 3958  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  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-1o 6279  df-2o 6280  df-er 6395  df-en 6601
This theorem is referenced by:  pr2nelem  7013  dju1p1e2  7017
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