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Theorem pm110.643 7803
Description: 1+1=2 for cardinal number addition, derived from pm54.43 7633 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 7565), but after applying definitions, our theorem is equivalent. The comment for cdaval 7796 explains why we use  ~~ instead of =. See pm110.643ALT 7804 for a shorter proof that doesn't use pm54.43 7633. (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 6486 . . 3  |-  1o  e.  On
2 cdaval 7796 . . 3  |-  ( ( 1o  e.  On  /\  1o  e.  On )  -> 
( 1o  +c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
31, 1, 2mp2an 653 . 2  |-  ( 1o 
+c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
4 xp01disj 6495 . . 3  |-  ( ( 1o  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
51elexi 2797 . . . . 5  |-  1o  e.  _V
6 0ex 4150 . . . . 5  |-  (/)  e.  _V
75, 6xpsnen 6946 . . . 4  |-  ( 1o 
X.  { (/) } ) 
~~  1o
85, 5xpsnen 6946 . . . 4  |-  ( 1o 
X.  { 1o }
)  ~~  1o
9 pm54.43 7633 . . . 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 653 . . 3  |-  ( ( ( 1o  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  (
( 1o  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  ~~  2o )
114, 10mpbi 199 . 2  |-  ( ( 1o  X.  { (/) } )  u.  ( 1o 
X.  { 1o }
) )  ~~  2o
123, 11eqbrtri 4042 1  |-  ( 1o 
+c  1o )  ~~  2o
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023   Oncon0 4392    X. cxp 4687  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ~~ cen 6860    +c ccda 7793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-cda 7794
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