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Theorem pm110.643 8882
 Description: 1+1=2 for cardinal number addition, derived from pm54.43 8709 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 8641), but after applying definitions, our theorem is equivalent. The comment for cdaval 8875 explains why we use ≈ instead of =. See pm110.643ALT 8883 for a shorter proof that doesn't use pm54.43 8709. (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 7454 . . 3 1𝑜 ∈ On
2 cdaval 8875 . . 3 ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
31, 1, 2mp2an 704 . 2 (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4 xp01disj 7463 . . 3 ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
51elexi 3186 . . . . 5 1𝑜 ∈ V
6 0ex 4718 . . . . 5 ∅ ∈ V
75, 6xpsnen 7929 . . . 4 (1𝑜 × {∅}) ≈ 1𝑜
85, 5xpsnen 7929 . . . 4 (1𝑜 × {1𝑜}) ≈ 1𝑜
9 pm54.43 8709 . . . 4 (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜))
107, 8, 9mp2an 704 . . 3 (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)
114, 10mpbi 219 . 2 ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜
123, 11eqbrtri 4604 1 (1𝑜 +𝑐 1𝑜) ≈ 2𝑜
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  1𝑜c1o 7440  2𝑜c2o 7441   ≈ cen 7838   +𝑐 ccda 8872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-cda 8873 This theorem is referenced by: (None)
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