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Theorem pm54.43 7121
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 7089), 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 7120. 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 7272 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 5947 . . . . . . . 8  |-  1o  e.  On
21elexi 2485 . . . . . . 7  |-  1o  e.  _V
32ensn1 6368 . . . . . 6  |-  { 1o }  ~~  1o
43ensymi 6355 . . . . 5  |-  1o  ~~  { 1o }
5 entr 6356 . . . . 5  |-  ( ( B  ~~  1o  /\  1o  ~~  { 1o }
)  ->  B  ~~  { 1o } )
64, 5mpan2 645 . . . 4  |-  ( B 
~~  1o  ->  B  ~~  { 1o } )
71onirri 4054 . . . . . . 7  |-  -.  1o  e.  1o
8 disjsn 3309 . . . . . . 7  |-  ( ( 1o  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  1o )
97, 8mpbir 198 . . . . . 6  |-  ( 1o 
i^i  { 1o } )  =  (/)
10 unen 6386 . . . . . 6  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  (
( A  i^i  B
)  =  (/)  /\  ( 1o  i^i  { 1o }
)  =  (/) ) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
119, 10mpanr2 658 . . . . 5  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) )
1211ex 420 . . . 4  |-  ( ( A  ~~  1o  /\  B  ~~  { 1o }
)  ->  ( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) ) )
136, 12sylan2 454 . . 3  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) ) )
14 df-2o 5941 . . . . 5  |-  2o  =  suc  1o
15 df-suc 3955 . . . . 5  |-  suc  1o  =  ( 1o  u.  { 1o } )
1614, 15eqtri 2085 . . . 4  |-  2o  =  ( 1o  u.  { 1o } )
1716breq2i 3600 . . 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 6371 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
20 en1 6371 . . 3  |-  ( B 
~~  1o  <->  E. y  B  =  { y } )
21 unidm 2941 . . . . . . . . . . . . . 14  |-  ( { x }  u.  {
x } )  =  { x }
22 sneq 3267 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  { x }  =  { y } )
2322uneq2d 2952 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( { x }  u.  { x } )  =  ( { x }  u.  { y } ) )
2421, 23syl5reqr 2112 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( { x }  u.  { y } )  =  { x } )
25 vex 2479 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
2625ensn1 6368 . . . . . . . . . . . . . 14  |-  { x }  ~~  1o
27 1sdom2 6551 . . . . . . . . . . . . . 14  |-  1o  ~<  2o
28 ensdomtr 6440 . . . . . . . . . . . . . 14  |-  ( ( { x }  ~~  1o  /\  1o  ~<  2o )  ->  { x }  ~<  2o )
2926, 27, 28mp2an 646 . . . . . . . . . . . . 13  |-  { x }  ~<  2o
3024, 29syl6eqbr 3629 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( { x }  u.  { y } )  ~<  2o )
31 sdomnen 6334 . . . . . . . . . . . 12  |-  ( ( { x }  u.  { y } )  ~<  2o  ->  -.  ( {
x }  u.  {
y } )  ~~  2o )
3230, 31syl 15 . . . . . . . . . . 11  |-  ( x  =  y  ->  -.  ( { x }  u.  { y } )  ~~  2o )
3332necon2ai 2227 . . . . . . . . . 10  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  x  =/=  y
)
34 disjsn2 3310 . . . . . . . . . 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 2947 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  u.  B
)  =  ( { x }  u.  {
y } ) )
3837breq1d 3602 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  <->  ( { x }  u.  { y } )  ~~  2o ) )
39 ineq12 2986 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  i^i  B
)  =  ( { x }  i^i  {
y } ) )
4039eqeq1d 2073 . . . . . . . 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 420 . . . . . 6  |-  ( A  =  { x }  ->  ( B  =  {
y }  ->  (
( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4342exlimdv 1782 . . . . 5  |-  ( A  =  { x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4443exlimiv 1868 . . . 4  |-  ( E. x  A  =  {
x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4544imp 415 . . 3  |-  ( ( E. x  A  =  { x }  /\  E. y  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4619, 20, 45syl2anb 459 . 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 1446    = wceq 1520    e. wcel 1522    =/= wne 2183    u. cun 2792    i^i cin 2793   (/)c0 3073   {csn 3256   class class class wbr 3592   Oncon0 3949   suc csuc 3951   1oc1o 5933   2oc2o 5934    ~~ cen 6304    ~< csdm 6306
This theorem is referenced by:  pr2nelem  7122  pm110.643  7272  isprm2lem  11886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1563  ax-10 1577  ax-9 1583  ax-4 1590  ax-16 1776  ax-ext 2047  ax-sep 3707  ax-nul 3715  ax-pow 3751  ax-pr 3775  ax-un 4067
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-tru 1259  df-ex 1447  df-sb 1737  df-eu 1959  df-mo 1960  df-clab 2053  df-cleq 2058  df-clel 2061  df-ne 2185  df-ral 2279  df-rex 2280  df-reu 2281  df-rab 2282  df-v 2478  df-sbc 2652  df-dif 2797  df-un 2799  df-in 2801  df-ss 2805  df-pss 2807  df-nul 3074  df-if 3183  df-pw 3244  df-sn 3262  df-pr 3263  df-tp 3264  df-op 3265  df-uni 3431  df-br 3593  df-opab 3647  df-tr 3680  df-eprel 3862  df-id 3866  df-po 3871  df-so 3872  df-fr 3909  df-we 3911  df-ord 3952  df-on 3953  df-lim 3954  df-suc 3955  df-om 4230  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-fun 4284  df-fn 4285  df-f 4286  df-f1 4287  df-fo 4288  df-f1o 4289  df-fv 4290  df-1o 5940  df-2o 5941  df-er 6121  df-en 6308  df-dom 6309  df-sdom 6310
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