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Theorem pm110.643 7798
Description: 1+1=2 for cardinal number addition, derived from pm54.43 7628 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 7560), but after applying definitions, our theorem is equivalent. The comment for cdaval 7791 explains why we use  ~~ instead of =. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
Assertion
Ref Expression
pm110.643  |-  ( 1o 
+c  1o )  ~~  2o

Proof of Theorem pm110.643
StepHypRef Expression
1 1on 6481 . . 3  |-  1o  e.  On
2 cdaval 7791 . . 3  |-  ( ( 1o  e.  On  /\  1o  e.  On )  -> 
( 1o  +c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
31, 1, 2mp2an 655 . 2  |-  ( 1o 
+c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
4 xp01disj 6490 . . 3  |-  ( ( 1o  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
51elexi 2798 . . . . 5  |-  1o  e.  _V
6 0ex 4151 . . . . 5  |-  (/)  e.  _V
75, 6xpsnen 6941 . . . 4  |-  ( 1o 
X.  { (/) } ) 
~~  1o
85, 5xpsnen 6941 . . . 4  |-  ( 1o 
X.  { 1o }
)  ~~  1o
9 pm54.43 7628 . . . 4  |-  ( ( ( 1o  X.  { (/)
} )  ~~  1o  /\  ( 1o  X.  { 1o } )  ~~  1o )  ->  ( ( ( 1o  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)  <->  (
( 1o  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  ~~  2o ) )
107, 8, 9mp2an 655 . . 3  |-  ( ( ( 1o  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  (
( 1o  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  ~~  2o )
114, 10mpbi 201 . 2  |-  ( ( 1o  X.  { (/) } )  u.  ( 1o 
X.  { 1o }
) )  ~~  2o
123, 11eqbrtri 4043 1  |-  ( 1o 
+c  1o )  ~~  2o
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1624    e. wcel 1685    u. cun 3151    i^i cin 3152   (/)c0 3456   {csn 3641   class class class wbr 4024   Oncon0 4391    X. cxp 4686  (class class class)co 5819   1oc1o 6467   2oc2o 6468    ~~ cen 6855    +c ccda 7788
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1o 6474  df-2o 6475  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-cda 7789
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