Theorem pm110.643 8578

Description: 1+1=2 for cardinal number addition, derived from pm54.43 8402 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 8334), but after applying definitions, our theorem is equivalent. The comment for cdaval 8571 explains why we use instead of =. See pm110.643ALT 8579 for a shorter proof that doesn't use pm54.43 8402. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)

Assertion

Ref	Expression
pm110.643

Proof of Theorem pm110.643

Step	Hyp	Ref	Expression
1		1on 7156	. . 3
2		cdaval 8571	. . 3
3	1, 1, 2	mp2an 672	. 2
4		xp01disj 7165	. . 3
5	1	elexi 3119	. . . . 5
6		0ex 4582	. . . . 5
7	5, 6	xpsnen 7621	. . . 4
8	5, 5	xpsnen 7621	. . . 4
9		pm54.43 8402	. . . 4
10	7, 8, 9	mp2an 672	. . 3
11	4, 10	mpbi 208	. 2
12	3, 11	eqbrtri 4471	1

Syntax hints:  <->wb 184  =wceq 1395  e.wcel 1818  u.cun 3473  i^icin 3474  c0 3784  {csn 4029   class class class wbr 4452  con0 4883  X.cxp 5002  (class class class)co 6296  c1o 7142  c2o 7143  cen 7533  ccda 8568

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-int 4287  df-br 4453  df-opab 4511  df-mpt 4512  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6701  df-1o 7149  df-2o 7150  df-er 7330  df-en 7537  df-dom 7538  df-sdom 7539  df-cda 8569