Theorem pm110.643 4895

Description: 1+1=2 for cardinal number addition. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Unlike us, 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 4698), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4546. The comment for cdaval 4892 explains why we use ~~ instead of =.

Assertion

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

Proof of Theorem pm110.643

Step	Hyp	Ref	Expression
1		1on 4122	. . . 4 |- 1o e. On
2	1	elisseti 1809	. . 3 |- 1o e. V
3	2, 2	cdaval 4892	. 2 |- (1o +c 1o) = ((1o X. {(/)}) u. (1o X. {1o}))
4		xp01disj 4127	. . 3 |- ((1o X. {(/)}) i^i (1o X. {1o})) = (/)
5		0ex 2701	. . . . 5 |- (/) e. V
6	2, 5	xpsnen 4415	. . . 4 |- (1o X. {(/)}) ~~ 1o
7	2, 2	xpsnen 4415	. . . 4 |- (1o X. {1o}) ~~ 1o
8		pm54.43 4546	. . . 4 |- (((1o X. {(/)}) ~~ 1o /\ (1o X. {1o}) ~~ 1o) -> (((1o X. {(/)}) i^i (1o X. {1o})) = (/) <-> ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o))
9	6, 7, 8	mp2an 695	. . 3 |- (((1o X. {(/)}) i^i (1o X. {1o})) = (/) <-> ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o)
10	4, 9	mpbi 189	. 2 |- ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o
11	3, 10	eqbrtr 2624	1 |- (1o +c 1o) ~~ 2o

Syntax hints:   <-> wb 146   = wceq 953   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609  Oncon0 2938   X. cxp 3158  (class class class)co 3948  1oc1o 4112  2oc2o 4113   ~~ cen 4348   +c ccda 4889

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-oprab 3951  df-1o 4117  df-2o 4118  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-cda 4890

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