HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Theorem pm54.43 6303
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 6280), so that their means, in our notation, which is the same as by pm54.43lem 6302. 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 6397 shows the derivation of 1+1=2 for cardinal numbers from this theorem.

Assertion
Ref Expression
pm54.43

Proof of Theorem pm54.43
StepHypRef Expression
1 1on 5420 . . . . . . 7
21elexi 2344 . . . . . 6
32ensn1 5754 . . . . . 6
42, 3ensymi 5743 . . . . 5
5 entr 5744 . . . . 5
64, 5mpan2 651 . . . 4
71onirri 3786 . . . . . . 7
8 disjsn 3122 . . . . . . 7
97, 8mpbir 198 . . . . . 6
10 unen 5769 . . . . . 6
119, 10mpanr2 664 . . . . 5
1211ex 423 . . . 4
136, 12sylan2 457 . . 3
14 df-2o 5414 . . . . 5
15 df-suc 3688 . . . . 5
1614, 15eqtri 1949 . . . 4
1716breq2i 3370 . . 3
1813, 17syl6ibr 217 . 2
19 en1 5757 . . 3
20 en1 5757 . . 3
21 unidm 2778 . . . . . . . . . . . . . 14
22 sneq 3089 . . . . . . . . . . . . . . 15
2322uneq2d 2789 . . . . . . . . . . . . . 14
2421, 23syl5reqr 1976 . . . . . . . . . . . . 13
25 vex 2338 . . . . . . . . . . . . . . 15
2625ensn1 5754 . . . . . . . . . . . . . 14
27 1sdom2 5921 . . . . . . . . . . . . . 14
28 ensdomtr 5809 . . . . . . . . . . . . . 14
2926, 27, 28mp2an 652 . . . . . . . . . . . . 13
3024, 29syl6eqbr 3399 . . . . . . . . . . . 12
31 sdomnen 5714 . . . . . . . . . . . 12
3230, 31syl 14 . . . . . . . . . . 11
3332necon2ai 2090 . . . . . . . . . 10
34 disjsn2 3124 . . . . . . . . . 10
3533, 34syl 14 . . . . . . . . 9
3635a1i 10 . . . . . . . 8
37 uneq12 2784 . . . . . . . . 9
3837breq1d 3372 . . . . . . . 8
39 ineq12 2821 . . . . . . . . 9
4039eqeq1d 1937 . . . . . . . 8
4136, 38, 403imtr4d 258 . . . . . . 7
4241ex 423 . . . . . 6
4342exlimdv 1646 . . . . 5
4443exlimiv 1729 . . . 4
4544imp 418 . . 3
4619, 20, 45syl2anb 462 . 2
4718, 46impbid 181 1
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wb 174   wa 360  wex 1346   wceq 1425   wcel 1427   wne 2047   cun 2636   cin 2637  c0 2903  csn 3078   class class class wbr 3362  con0 3682   csuc 3684  c1o 5406  c2o 5407   cen 5691   csdm 5693
This theorem is referenced by:  pr2nelem 6304  pm110.643 6397  isprm2lem 9754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1342  ax-6 1343  ax-7 1344  ax-gen 1345  ax-8 1429  ax-10 1430  ax-11 1431  ax-12 1432  ax-13 1433  ax-14 1434  ax-17 1441  ax-9 1456  ax-4 1462  ax-16 1640  ax-ext 1911  ax-rep 3448  ax-sep 3458  ax-nul 3467  ax-pow 3503  ax-pr 3527  ax-un 3799
This theorem depends on definitions:  df-bi 175  df-or 361  df-an 362  df-3or 913  df-3an 914  df-tru 1320  df-ex 1347  df-sb 1602  df-eu 1829  df-mo 1830  df-clab 1917  df-cleq 1922  df-clel 1925  df-ne 2049  df-ral 2142  df-rex 2143  df-reu 2144  df-rab 2145  df-v 2337  df-dif 2641  df-un 2643  df-in 2645  df-ss 2647  df-pss 2649  df-nul 2904  df-pw 3067  df-sn 3084  df-pr 3085  df-tp 3086  df-op 3087  df-uni 3218  df-br 3363  df-opab 3417  df-tr 3432  df-eprel 3612  df-id 3615  df-po 3620  df-so 3634  df-fr 3653  df-we 3669  df-ord 3685  df-on 3686  df-suc 3688  df-xp 4009  df-rel 4010  df-cnv 4011  df-co 4012  df-dm 4013  df-rn 4014  df-res 4015  df-ima 4016  df-fun 4017  df-fn 4018  df-f 4019  df-f1 4020  df-fo 4021  df-f1o 4022  df-fv 4023  df-1o 5413  df-2o 5414  df-er 5550  df-en 5695  df-dom 5696  df-sdom 5697
Copyright terms: Public domain