Theorem pm110.643 5074

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 4872), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4715. The comment for cdavali 5070 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 4274	. . . 4 |- 1o e. On
2	1	elisseti 1864	. . 3 |- 1o e. V
3	2, 2	cdavali 5070	. 2 |- (1o +c 1o) = ((1o X. {(/)}) u. (1o X. {1o}))
4		xp01disj 4279	. . 3 |- ((1o X. {(/)}) i^i (1o X. {1o})) = (/)
5		0ex 2785	. . . . 5 |- (/) e. V
6	2, 5	xpsnen 4576	. . . 4 |- (1o X. {(/)}) ~~ 1o
7	2, 2	xpsnen 4576	. . . 4 |- (1o X. {1o}) ~~ 1o
8		pm54.43 4715	. . . 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 701	. . 3 |- (((1o X. {(/)}) i^i (1o X. {1o})) = (/) <-> ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o)
10	4, 9	mpbi 187	. 2 |- ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o
11	3, 10	eqbrtri 2707	1 |- (1o +c 1o) ~~ 2o

Syntax hints:   <-> wb 144   = wceq 992   u. cun 2097   i^i cin 2098  (/)c0 2332  {csn 2467   class class class wbr 2692  Oncon0 2975   X. cxp 3249  (class class class)co 4021  1oc1o 4264  2oc2o 4265   ~~ cen 4505   +c ccda 5067

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1o 4269  df-2o 4270  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-cda 5068

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