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Theorem pm110.643 7757
Description: 1+1=2 for cardinal number addition, derived from pm54.43 7587 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 7519), but after applying definitions, our theorem is equivalent. The comment for cdaval 7750 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 6440 . . 3  |-  1o  e.  On
2 cdaval 7750 . . 3  |-  ( ( 1o  e.  On  /\  1o  e.  On )  -> 
( 1o  +c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
31, 1, 2mp2an 656 . 2  |-  ( 1o 
+c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
4 xp01disj 6449 . . 3  |-  ( ( 1o  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
51elexi 2766 . . . . 5  |-  1o  e.  _V
6 0ex 4110 . . . . 5  |-  (/)  e.  _V
75, 6xpsnen 6900 . . . 4  |-  ( 1o 
X.  { (/) } ) 
~~  1o
85, 5xpsnen 6900 . . . 4  |-  ( 1o 
X.  { 1o }
)  ~~  1o
9 pm54.43 7587 . . . 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 656 . . 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 4002 1  |-  ( 1o 
+c  1o )  ~~  2o
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621    u. cun 3111    i^i cin 3112   (/)c0 3416   {csn 3600   class class class wbr 3983   Oncon0 4350    X. cxp 4645  (class class class)co 5778   1oc1o 6426   2oc2o 6427    ~~ cen 6814    +c ccda 7747
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1o 6433  df-2o 6434  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-cda 7748
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