Theorem pm110.643 4906

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 4709), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4555. The comment for cdaval 4903 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 4131	. . . 4 |- 1o e. On
2	1	elisseti 1815	. . 3 |- 1o e. V
3	2, 2	cdaval 4903	. 2 |- (1o +c 1o) = ((1o X. {(/)}) u. (1o X. {1o}))
4		xp01disj 4136	. . 3 |- ((1o X. {(/)}) i^i (1o X. {1o})) = (/)
5		0ex 2707	. . . . 5 |- (/) e. V
6	2, 5	xpsnen 4424	. . . 4 |- (1o X. {(/)}) ~~ 1o
7	2, 2	xpsnen 4424	. . . 4 |- (1o X. {1o}) ~~ 1o
8		pm54.43 4555	. . . 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 696	. . 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 2630	1 |- (1o +c 1o) ~~ 2o

Syntax hints:   <-> wb 146   = wceq 955   u. cun 2042   i^i cin 2043  (/)c0 2277  {csn 2406   class class class wbr 2615  Oncon0 2944   X. cxp 3164  (class class class)co 3958  1oc1o 4121  2oc2o 4122   ~~ cen 4357   +c ccda 4900

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-opr 3960  df-oprab 3961  df-1o 4126  df-2o 4127  df-er 4254  df-en 4360  df-dom 4361  df-sdom 4362  df-cda 4901

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