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Theorem pm54.43 8402
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 8370), so that their means, in our notation, which is the same as by pm54.43lem 8401. 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 8578 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

Assertion
Ref Expression
pm54.43

Proof of Theorem pm54.43
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 7156 . . . . . . . 8
21elexi 3119 . . . . . . 7
32ensn1 7599 . . . . . 6
43ensymi 7585 . . . . 5
5 entr 7587 . . . . 5
64, 5mpan2 671 . . . 4
71onirri 4989 . . . . . . 7
8 disjsn 4090 . . . . . . 7
97, 8mpbir 209 . . . . . 6
10 unen 7618 . . . . . 6
119, 10mpanr2 684 . . . . 5
1211ex 434 . . . 4
136, 12sylan2 474 . . 3
14 df-2o 7150 . . . . 5
15 df-suc 4889 . . . . 5
1614, 15eqtri 2486 . . . 4
1716breq2i 4460 . . 3
1813, 17syl6ibr 227 . 2
19 en1 7602 . . 3
20 en1 7602 . . 3
21 unidm 3646 . . . . . . . . . . . . . 14
22 sneq 4039 . . . . . . . . . . . . . . 15
2322uneq2d 3657 . . . . . . . . . . . . . 14
2421, 23syl5reqr 2513 . . . . . . . . . . . . 13
25 vex 3112 . . . . . . . . . . . . . . 15
2625ensn1 7599 . . . . . . . . . . . . . 14
27 1sdom2 7738 . . . . . . . . . . . . . 14
28 ensdomtr 7673 . . . . . . . . . . . . . 14
2926, 27, 28mp2an 672 . . . . . . . . . . . . 13
3024, 29syl6eqbr 4489 . . . . . . . . . . . 12
31 sdomnen 7564 . . . . . . . . . . . 12
3230, 31syl 16 . . . . . . . . . . 11
3332necon2ai 2692 . . . . . . . . . 10
34 disjsn2 4091 . . . . . . . . . 10
3533, 34syl 16 . . . . . . . . 9
3635a1i 11 . . . . . . . 8
37 uneq12 3652 . . . . . . . . 9
3837breq1d 4462 . . . . . . . 8
39 ineq12 3694 . . . . . . . . 9
4039eqeq1d 2459 . . . . . . . 8
4136, 38, 403imtr4d 268 . . . . . . 7
4241ex 434 . . . . . 6
4342exlimdv 1724 . . . . 5
4443exlimiv 1722 . . . 4
4544imp 429 . . 3
4619, 20, 45syl2anb 479 . 2
4718, 46impbid 191 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652  u.cun 3473  i^icin 3474   c0 3784  {csn 4029   class class class wbr 4452   con0 4883  succsuc 4885   c1o 7142   c2o 7143   cen 7533   csdm 7535
This theorem is referenced by:  pr2nelem  8403  pm110.643  8578  isprm2lem  14224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6701  df-1o 7149  df-2o 7150  df-er 7330  df-en 7537  df-dom 7538  df-sdom 7539
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