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Theorem pm54.43 7105
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 7073), 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 7104. 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 7256 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
StepHypRef Expression
1 1on 5933 . . . . . . . 8  |-  1o  e.  On
21elexi 2481 . . . . . . 7  |-  1o  e.  _V
32ensn1 6354 . . . . . 6  |-  { 1o }  ~~  1o
43ensymi 6341 . . . . 5  |-  1o  ~~  { 1o }
5 entr 6342 . . . . 5  |-  ( ( B  ~~  1o  /\  1o  ~~  { 1o }
)  ->  B  ~~  { 1o } )
64, 5mpan2 643 . . . 4  |-  ( B 
~~  1o  ->  B  ~~  { 1o } )
71onirri 4044 . . . . . . 7  |-  -.  1o  e.  1o
8 disjsn 3304 . . . . . . 7  |-  ( ( 1o  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  1o )
97, 8mpbir 198 . . . . . 6  |-  ( 1o 
i^i  { 1o } )  =  (/)
10 unen 6372 . . . . . 6  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  (
( A  i^i  B
)  =  (/)  /\  ( 1o  i^i  { 1o }
)  =  (/) ) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
119, 10mpanr2 656 . . . . 5  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) )
1211ex 419 . . . 4  |-  ( ( A  ~~  1o  /\  B  ~~  { 1o }
)  ->  ( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) ) )
136, 12sylan2 453 . . 3  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) ) )
14 df-2o 5927 . . . . 5  |-  2o  =  suc  1o
15 df-suc 3945 . . . . 5  |-  suc  1o  =  ( 1o  u.  { 1o } )
1614, 15eqtri 2081 . . . 4  |-  2o  =  ( 1o  u.  { 1o } )
1716breq2i 3590 . . 3  |-  ( ( A  u.  B ) 
~~  2o  <->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
1813, 17syl6ibr 216 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  2o ) )
19 en1 6357 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
20 en1 6357 . . 3  |-  ( B 
~~  1o  <->  E. y  B  =  { y } )
21 unidm 2937 . . . . . . . . . . . . . 14  |-  ( { x }  u.  {
x } )  =  { x }
22 sneq 3262 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  { x }  =  { y } )
2322uneq2d 2948 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( { x }  u.  { x } )  =  ( { x }  u.  { y } ) )
2421, 23syl5reqr 2108 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( { x }  u.  { y } )  =  { x } )
25 vex 2475 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
2625ensn1 6354 . . . . . . . . . . . . . 14  |-  { x }  ~~  1o
27 1sdom2 6537 . . . . . . . . . . . . . 14  |-  1o  ~<  2o
28 ensdomtr 6426 . . . . . . . . . . . . . 14  |-  ( ( { x }  ~~  1o  /\  1o  ~<  2o )  ->  { x }  ~<  2o )
2926, 27, 28mp2an 644 . . . . . . . . . . . . 13  |-  { x }  ~<  2o
3024, 29syl6eqbr 3619 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( { x }  u.  { y } )  ~<  2o )
31 sdomnen 6320 . . . . . . . . . . . 12  |-  ( ( { x }  u.  { y } )  ~<  2o  ->  -.  ( {
x }  u.  {
y } )  ~~  2o )
3230, 31syl 15 . . . . . . . . . . 11  |-  ( x  =  y  ->  -.  ( { x }  u.  { y } )  ~~  2o )
3332necon2ai 2223 . . . . . . . . . 10  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  x  =/=  y
)
34 disjsn2 3305 . . . . . . . . . 10  |-  ( x  =/=  y  ->  ( { x }  i^i  { y } )  =  (/) )
3533, 34syl 15 . . . . . . . . 9  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) )
3635a1i 10 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) ) )
37 uneq12 2943 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  u.  B
)  =  ( { x }  u.  {
y } ) )
3837breq1d 3592 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  <->  ( { x }  u.  { y } )  ~~  2o ) )
39 ineq12 2982 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  i^i  B
)  =  ( { x }  i^i  {
y } ) )
4039eqeq1d 2069 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  i^i  B )  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
4136, 38, 403imtr4d 257 . . . . . . 7  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4241ex 419 . . . . . 6  |-  ( A  =  { x }  ->  ( B  =  {
y }  ->  (
( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4342exlimdv 1778 . . . . 5  |-  ( A  =  { x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4443exlimiv 1864 . . . 4  |-  ( E. x  A  =  {
x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4544imp 414 . . 3  |-  ( ( E. x  A  =  { x }  /\  E. y  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4619, 20, 45syl2anb 458 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4718, 46impbid 181 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 174    /\ wa 356   E.wex 1443    = wceq 1517    e. wcel 1519    =/= wne 2179    u. cun 2788    i^i cin 2789   (/)c0 3069   {csn 3251   class class class wbr 3582   Oncon0 3939   suc csuc 3941   1oc1o 5919   2oc2o 5920    ~~ cen 6290    ~< csdm 6292
This theorem is referenced by:  pr2nelem  7106  pm110.643  7256  isprm2lem  11279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-13 1523  ax-14 1524  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043  ax-sep 3697  ax-nul 3705  ax-pow 3741  ax-pr 3765  ax-un 4057
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 894  df-3an 895  df-tru 1256  df-ex 1444  df-sb 1733  df-eu 1955  df-mo 1956  df-clab 2049  df-cleq 2054  df-clel 2057  df-ne 2181  df-ral 2275  df-rex 2276  df-reu 2277  df-rab 2278  df-v 2474  df-sbc 2648  df-dif 2793  df-un 2795  df-in 2797  df-ss 2801  df-pss 2803  df-nul 3070  df-if 3178  df-pw 3239  df-sn 3257  df-pr 3258  df-tp 3259  df-op 3260  df-uni 3421  df-br 3583  df-opab 3637  df-tr 3670  df-eprel 3852  df-id 3856  df-po 3861  df-so 3862  df-fr 3899  df-we 3901  df-ord 3942  df-on 3943  df-lim 3944  df-suc 3945  df-om 4220  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-fun 4274  df-fn 4275  df-f 4276  df-f1 4277  df-fo 4278  df-f1o 4279  df-fv 4280  df-1o 5926  df-2o 5927  df-er 6107  df-en 6294  df-dom 6295  df-sdom 6296
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