MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm110.643 Unicode version

Theorem pm110.643 7684
Description: 1+1=2 for cardinal number addition, derived from pm54.43 7514 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 7446), but after applying definitions, our theorem is equivalent. The comment for cdaval 7677 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 6369 . . 3  |-  1o  e.  On
2 cdaval 7677 . . 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 6378 . . 3  |-  ( ( 1o  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
51elexi 2734 . . . . 5  |-  1o  e.  _V
6 0ex 4044 . . . . 5  |-  (/)  e.  _V
75, 6xpsnen 6828 . . . 4  |-  ( 1o 
X.  { (/) } ) 
~~  1o
85, 5xpsnen 6828 . . . 4  |-  ( 1o 
X.  { 1o }
)  ~~  1o
9 pm54.43 7514 . . . 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 3936 1  |-  ( 1o 
+c  1o )  ~~  2o
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621    u. cun 3073    i^i cin 3074   (/)c0 3359   {csn 3541   class class class wbr 3917   Oncon0 4282    X. cxp 4575  (class class class)co 5707   1oc1o 6355   2oc2o 6356    ~~ cen 6743    +c ccda 7674
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 4035  ax-nul 4043  ax-pow 4079  ax-pr 4105  ax-un 4400
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 2511  df-rex 2512  df-reu 2513  df-rab 2514  df-v 2727  df-sbc 2920  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-pss 3088  df-nul 3360  df-if 3468  df-pw 3529  df-sn 3547  df-pr 3548  df-tp 3549  df-op 3550  df-uni 3725  df-int 3758  df-br 3918  df-opab 3972  df-mpt 3973  df-tr 4008  df-eprel 4195  df-id 4199  df-po 4204  df-so 4205  df-fr 4242  df-we 4244  df-ord 4285  df-on 4286  df-lim 4287  df-suc 4288  df-om 4545  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 4599  df-fn 4600  df-f 4601  df-f1 4602  df-fo 4603  df-f1o 4604  df-fv 4605  df-ov 5710  df-oprab 5711  df-mpt2 5712  df-1o 6362  df-2o 6363  df-er 6543  df-en 6747  df-dom 6748  df-sdom 6749  df-cda 7675
  Copyright terms: Public domain W3C validator