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Theorem pm54.43 7517
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 7485), 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 7516. 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 7687 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 6372 . . . . . . . 8  |-  1o  e.  On
21elexi 2736 . . . . . . 7  |-  1o  e.  _V
32ensn1 6810 . . . . . 6  |-  { 1o }  ~~  1o
43ensymi 6797 . . . . 5  |-  1o  ~~  { 1o }
5 entr 6798 . . . . 5  |-  ( ( B  ~~  1o  /\  1o  ~~  { 1o }
)  ->  B  ~~  { 1o } )
64, 5mpan2 655 . . . 4  |-  ( B 
~~  1o  ->  B  ~~  { 1o } )
71onirri 4390 . . . . . . 7  |-  -.  1o  e.  1o
8 disjsn 3597 . . . . . . 7  |-  ( ( 1o  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  1o )
97, 8mpbir 202 . . . . . 6  |-  ( 1o 
i^i  { 1o } )  =  (/)
10 unen 6828 . . . . . 6  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  (
( A  i^i  B
)  =  (/)  /\  ( 1o  i^i  { 1o }
)  =  (/) ) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
119, 10mpanr2 668 . . . . 5  |-  ( ( ( A  ~~  1o  /\  B  ~~  { 1o } )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) )
1211ex 425 . . . 4  |-  ( ( A  ~~  1o  /\  B  ~~  { 1o }
)  ->  ( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) ) )
136, 12sylan2 462 . . 3  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  ( 1o  u.  { 1o } ) ) )
14 df-2o 6366 . . . . 5  |-  2o  =  suc  1o
15 df-suc 4291 . . . . 5  |-  suc  1o  =  ( 1o  u.  { 1o } )
1614, 15eqtri 2273 . . . 4  |-  2o  =  ( 1o  u.  { 1o } )
1716breq2i 3928 . . 3  |-  ( ( A  u.  B ) 
~~  2o  <->  ( A  u.  B )  ~~  ( 1o  u.  { 1o }
) )
1813, 17syl6ibr 220 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  ->  ( A  u.  B )  ~~  2o ) )
19 en1 6813 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
20 en1 6813 . . 3  |-  ( B 
~~  1o  <->  E. y  B  =  { y } )
21 unidm 3228 . . . . . . . . . . . . . 14  |-  ( { x }  u.  {
x } )  =  { x }
22 sneq 3555 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  { x }  =  { y } )
2322uneq2d 3239 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( { x }  u.  { x } )  =  ( { x }  u.  { y } ) )
2421, 23syl5reqr 2300 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( { x }  u.  { y } )  =  { x } )
25 vex 2730 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
2625ensn1 6810 . . . . . . . . . . . . . 14  |-  { x }  ~~  1o
27 1sdom2 6946 . . . . . . . . . . . . . 14  |-  1o  ~<  2o
28 ensdomtr 6882 . . . . . . . . . . . . . 14  |-  ( ( { x }  ~~  1o  /\  1o  ~<  2o )  ->  { x }  ~<  2o )
2926, 27, 28mp2an 656 . . . . . . . . . . . . 13  |-  { x }  ~<  2o
3024, 29syl6eqbr 3957 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( { x }  u.  { y } )  ~<  2o )
31 sdomnen 6776 . . . . . . . . . . . 12  |-  ( ( { x }  u.  { y } )  ~<  2o  ->  -.  ( {
x }  u.  {
y } )  ~~  2o )
3230, 31syl 17 . . . . . . . . . . 11  |-  ( x  =  y  ->  -.  ( { x }  u.  { y } )  ~~  2o )
3332necon2ai 2457 . . . . . . . . . 10  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  x  =/=  y
)
34 disjsn2 3598 . . . . . . . . . 10  |-  ( x  =/=  y  ->  ( { x }  i^i  { y } )  =  (/) )
3533, 34syl 17 . . . . . . . . 9  |-  ( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) )
3635a1i 12 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( { x }  u.  { y } )  ~~  2o  ->  ( { x }  i^i  { y } )  =  (/) ) )
37 uneq12 3234 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  u.  B
)  =  ( { x }  u.  {
y } ) )
3837breq1d 3930 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  <->  ( { x }  u.  { y } )  ~~  2o ) )
39 ineq12 3273 . . . . . . . . 9  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( A  i^i  B
)  =  ( { x }  i^i  {
y } ) )
4039eqeq1d 2261 . . . . . . . 8  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  i^i  B )  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
4136, 38, 403imtr4d 261 . . . . . . 7  |-  ( ( A  =  { x }  /\  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4241ex 425 . . . . . 6  |-  ( A  =  { x }  ->  ( B  =  {
y }  ->  (
( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4342exlimdv 1932 . . . . 5  |-  ( A  =  { x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4443exlimiv 2023 . . . 4  |-  ( E. x  A  =  {
x }  ->  ( E. y  B  =  { y }  ->  ( ( A  u.  B
)  ~~  2o  ->  ( A  i^i  B )  =  (/) ) ) )
4544imp 420 . . 3  |-  ( ( E. x  A  =  { x }  /\  E. y  B  =  {
y } )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4619, 20, 45syl2anb 467 . 2  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  u.  B )  ~~  2o  ->  ( A  i^i  B
)  =  (/) ) )
4718, 46impbid 185 1  |-  ( ( A  ~~  1o  /\  B  ~~  1o )  -> 
( ( A  i^i  B )  =  (/)  <->  ( A  u.  B )  ~~  2o ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2412    u. cun 3076    i^i cin 3077   (/)c0 3362   {csn 3544   class class class wbr 3920   Oncon0 4285   suc csuc 4287   1oc1o 6358   2oc2o 6359    ~~ cen 6746    ~< csdm 6748
This theorem is referenced by:  pr2nelem  7518  pm110.643  7687  isprm2lem  12639
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-1o 6365  df-2o 6366  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752
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