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Theorem pm110.643 8757
Description: 1+1=2 for cardinal number addition, derived from pm54.43 8584 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 8516), but after applying definitions, our theorem is equivalent. The comment for cdaval 8750 explains why we use instead of =. See pm110.643ALT 8758 for a shorter proof that doesn't use pm54.43 8584. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
Assertion
Ref Expression
pm110.643 (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Proof of Theorem pm110.643
StepHypRef Expression
1 1on 7329 . . 3 1𝑜 ∈ On
2 cdaval 8750 . . 3 ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
31, 1, 2mp2an 703 . 2 (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4 xp01disj 7338 . . 3 ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
51elexi 3090 . . . . 5 1𝑜 ∈ V
6 0ex 4617 . . . . 5 ∅ ∈ V
75, 6xpsnen 7804 . . . 4 (1𝑜 × {∅}) ≈ 1𝑜
85, 5xpsnen 7804 . . . 4 (1𝑜 × {1𝑜}) ≈ 1𝑜
9 pm54.43 8584 . . . 4 (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜))
107, 8, 9mp2an 703 . . 3 (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)
114, 10mpbi 218 . 2 ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜
123, 11eqbrtri 4502 1 (1𝑜 +𝑐 1𝑜) ≈ 2𝑜
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1938  cun 3442  cin 3443  c0 3777  {csn 4028   class class class wbr 4481   × cxp 4930  Oncon0 5530  (class class class)co 6425  1𝑜c1o 7315  2𝑜c2o 7316  cen 7713   +𝑐 ccda 8747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6833  df-1o 7322  df-2o 7323  df-er 7504  df-en 7717  df-dom 7718  df-sdom 7719  df-cda 8748
This theorem is referenced by: (None)
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