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Theorem pm54.43 7887
Description: Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2.

Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7855), so that their  A  e.  1 means, in our notation,  A  e.  { x  |  (
card `  x )  =  1o } which is the same as  A  ~~  1o by pm54.43lem 7886. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.)

Theorem pm110.643 8057 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (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 6731 . . . . . . . 8  |-  1o  e.  On
21elexi 2965 . . . . . . 7  |-  1o  e.  _V
32ensn1 7171 . . . . . 6  |-  { 1o }  ~~  1o
43ensymi 7157 . . . . 5  |-  1o  ~~  { 1o }
5 entr 7159 . . . . 5  |-  ( ( B  ~~  1o  /\  1o  ~~  { 1o }
)  ->  B  ~~  { 1o } )
64, 5mpan2 653 . . . 4  |-  ( B 
~~  1o  ->  B  ~~  { 1o } )
71onirri 4688 . . . . . . 7  |-  -.  1o  e.  1o
8 disjsn 3868 . . . . . . 7  |-  ( ( 1o  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  1o )
97, 8mpbir 201 . . . . . 6  |-  ( 1o 
i^i  { 1o } )  =  (/)
10 unen 7189 . . . . . 6  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  (
( A  i^i  B
)  =  (/)  /\  ( 1o  i^i  { 1o }
)  =  (/) ) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
119, 10mpanr2 666 . . . . 5  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) )
1211ex 424 . . . 4  |-  ( ( A  ~~  1o  /\  B  ~~  { 1o }
)  ->  ( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) ) )
136, 12sylan2 461 . . 3  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) ) )
14 df-2o 6725 . . . . 5  |-  2o  =  suc  1o
15 df-suc 4587 . . . . 5  |-  suc  1o  =  ( 1o  u.  { 1o } )
1614, 15eqtri 2456 . . . 4  |-  2o  =  ( 1o  u.  { 1o } )
1716breq2i 4220 . . 3  |-  ( ( A  u.  B ) 
~~  2o  <->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
1813, 17syl6ibr 219 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  2o ) )
19 en1 7174 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
20 en1 7174 . . 3  |-  ( B 
~~  1o  <->  E. y  B  =  { y } )
21 unidm 3490 . . . . . . . . . . . . . 14  |-  ( { x }  u.  {
x } )  =  { x }
22 sneq 3825 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  { x }  =  { y } )
2322uneq2d 3501 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( { x }  u.  { x } )  =  ( { x }  u.  { y } ) )
2421, 23syl5reqr 2483 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( { x }  u.  { y } )  =  { x } )
25 vex 2959 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
2625ensn1 7171 . . . . . . . . . . . . . 14  |-  { x }  ~~  1o
27 1sdom2 7307 . . . . . . . . . . . . . 14  |-  1o  ~<  2o
28 ensdomtr 7243 . . . . . . . . . . . . . 14  |-  ( ( { x }  ~~  1o  /\  1o  ~<  2o )  ->  { x }  ~<  2o )
2926, 27, 28mp2an 654 . . . . . . . . . . . . 13  |-  { x }  ~<  2o
3024, 29syl6eqbr 4249 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( { x }  u.  { y } )  ~<  2o )
31 sdomnen 7136 . . . . . . . . . . . 12  |-  ( ( { x }  u.  { y } )  ~<  2o  ->  -.  ( {
x }  u.  {
y } )  ~~  2o )
3230, 31syl 16 . . . . . . . . . . 11  |-  ( x  =  y  ->  -.  ( { x }  u.  { y } )  ~~  2o )
3332necon2ai 2649 . . . . . . . . . 10  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  x  =/=  y
)
34 disjsn2 3869 . . . . . . . . . 10  |-  ( x  =/=  y  ->  ( { x }  i^i  { y } )  =  (/) )
3533, 34syl 16 . . . . . . . . 9  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) )
3635a1i 11 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) ) )
37 uneq12 3496 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  u.  B
)  =  ( { x }  u.  {
y } ) )
3837breq1d 4222 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  <->  ( { x }  u.  { y } )  ~~  2o ) )
39 ineq12 3537 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  i^i  B
)  =  ( { x }  i^i  {
y } ) )
4039eqeq1d 2444 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  i^i  B )  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
4136, 38, 403imtr4d 260 . . . . . . 7  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4241ex 424 . . . . . 6  |-  ( A  =  { x }  ->  ( B  =  {
y }  ->  (
( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4342exlimdv 1646 . . . . 5  |-  ( A  =  { x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4443exlimiv 1644 . . . 4  |-  ( E. x  A  =  {
x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4544imp 419 . . 3  |-  ( ( E. x  A  =  { x }  /\  E. y  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4619, 20, 45syl2anb 466 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4718, 46impbid 184 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    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599    u. cun 3318    i^i cin 3319   (/)c0 3628   {csn 3814   class class class wbr 4212   Oncon0 4581   suc csuc 4583   1oc1o 6717   2oc2o 6718    ~~ cen 7106    ~< csdm 7108
This theorem is referenced by:  pr2nelem  7888  pm110.643  8057  isprm2lem  13086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-2o 6725  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112
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