Theorem pm110.643 8999

Description: 1+1=2 for cardinal number addition, derived from pm54.43 8826 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 8758), but after applying definitions, our theorem is equivalent. The comment for cdaval 8992 explains why we use ~~ instead of =. See pm110.643ALT 9000 for a shorter proof that doesn't use pm54.43 8826. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)

Assertion

Ref	Expression
pm110.643	|- ( 1o +c 1o ) ~~ 2o

Proof of Theorem pm110.643

Step	Hyp	Ref	Expression
1		1on 7567	. . 3 |- 1o e. On
2		cdaval 8992	. . 3 |- ( ( 1o e. On /\ 1o e. On ) -> ( 1o +c 1o ) = ( ( 1o X. { (/) } ) u. ( 1o X. { 1o } ) ) )
3	1, 1, 2	mp2an 708	. 2 |- ( 1o +c 1o ) = ( ( 1o X. { (/) } ) u. ( 1o X. { 1o } ) )
4		xp01disj 7576	. . 3 |- ( ( 1o X. { (/) } ) i^i ( 1o X. { 1o } ) ) = (/)
5	1	elexi 3213	. . . . 5 |- 1o e. _V
6		0ex 4790	. . . . 5 |- (/) e. _V
7	5, 6	xpsnen 8044	. . . 4 |- ( 1o X. { (/) } ) ~~ 1o
8	5, 5	xpsnen 8044	. . . 4 |- ( 1o X. { 1o } ) ~~ 1o
9		pm54.43 8826	. . . 4 |- ( ( ( 1o X. { (/) } ) ~~ 1o /\ ( 1o X. { 1o } ) ~~ 1o ) -> ( ( ( 1o X. { (/) } ) i^i ( 1o X. { 1o } ) ) = (/) <-> ( ( 1o X. { (/) } ) u. ( 1o X. { 1o } ) ) ~~ 2o ) )
10	7, 8, 9	mp2an 708	. . 3 |- ( ( ( 1o X. { (/) } ) i^i ( 1o X. { 1o } ) ) = (/) <-> ( ( 1o X. { (/) } ) u. ( 1o X. { 1o } ) ) ~~ 2o )
11	4, 10	mpbi 220	. 2 |- ( ( 1o X. { (/) } ) u. ( 1o X. { 1o } ) ) ~~ 2o
12	3, 11	eqbrtri 4674	1 |- ( 1o +c 1o ) ~~ 2o

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   Oncon0 5723  (class class class)co 6650   1oc1o 7553   2oc2o 7554   ~~ cen 7952   +c ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-cda 8990

This theorem is referenced by: (None)