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Theorem pm110.643 8859
Description: 1+1=2 for cardinal number addition, derived from pm54.43 8686 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 8618), but after applying definitions, our theorem is equivalent. The comment for cdaval 8852 explains why we use instead of =. See pm110.643ALT 8860 for a shorter proof that doesn't use pm54.43 8686. (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 7431 . . 3 1𝑜 ∈ On
2 cdaval 8852 . . 3 ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
31, 1, 2mp2an 703 . 2 (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4 xp01disj 7440 . . 3 ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
51elexi 3185 . . . . 5 1𝑜 ∈ V
6 0ex 4713 . . . . 5 ∅ ∈ V
75, 6xpsnen 7906 . . . 4 (1𝑜 × {∅}) ≈ 1𝑜
85, 5xpsnen 7906 . . . 4 (1𝑜 × {1𝑜}) ≈ 1𝑜
9 pm54.43 8686 . . . 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 4598 1 (1𝑜 +𝑐 1𝑜) ≈ 2𝑜
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1976  cun 3537  cin 3538  c0 3873  {csn 4124   class class class wbr 4577   × cxp 5026  Oncon0 5626  (class class class)co 6527  1𝑜c1o 7417  2𝑜c2o 7418  cen 7815   +𝑐 ccda 8849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
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 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1o 7424  df-2o 7425  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-cda 8850
This theorem is referenced by: (None)
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