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Theorem pm54.43 8589
Description: Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2.

Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 8557), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} which is the same as 𝐴 ≈ 1𝑜 by pm54.43lem 8588. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.)

Theorem pm110.643 8762 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

Assertion
Ref Expression
pm54.43 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2𝑜))

Proof of Theorem pm54.43
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 7334 . . . . . . . 8 1𝑜 ∈ On
21elexi 3090 . . . . . . 7 1𝑜 ∈ V
32ensn1 7786 . . . . . 6 {1𝑜} ≈ 1𝑜
43ensymi 7772 . . . . 5 1𝑜 ≈ {1𝑜}
5 entr 7774 . . . . 5 ((𝐵 ≈ 1𝑜 ∧ 1𝑜 ≈ {1𝑜}) → 𝐵 ≈ {1𝑜})
64, 5mpan2 702 . . . 4 (𝐵 ≈ 1𝑜𝐵 ≈ {1𝑜})
71onirri 5641 . . . . . . 7 ¬ 1𝑜 ∈ 1𝑜
8 disjsn 4095 . . . . . . 7 ((1𝑜 ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ 1𝑜)
97, 8mpbir 219 . . . . . 6 (1𝑜 ∩ {1𝑜}) = ∅
10 unen 7805 . . . . . 6 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ ((𝐴𝐵) = ∅ ∧ (1𝑜 ∩ {1𝑜}) = ∅)) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
119, 10mpanr2 715 . . . . 5 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1211ex 448 . . . 4 ((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
136, 12sylan2 489 . . 3 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
14 df-2o 7328 . . . . 5 2𝑜 = suc 1𝑜
15 df-suc 5536 . . . . 5 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
1614, 15eqtri 2536 . . . 4 2𝑜 = (1𝑜 ∪ {1𝑜})
1716breq2i 4489 . . 3 ((𝐴𝐵) ≈ 2𝑜 ↔ (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1813, 17syl6ibr 240 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2𝑜))
19 en1 7789 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 7789 . . 3 (𝐵 ≈ 1𝑜 ↔ ∃𝑦 𝐵 = {𝑦})
21 unidm 3622 . . . . . . . . . . . . . 14 ({𝑥} ∪ {𝑥}) = {𝑥}
22 sneq 4038 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2322uneq2d 3633 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
2421, 23syl5reqr 2563 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25 vex 3080 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2625ensn1 7786 . . . . . . . . . . . . . 14 {𝑥} ≈ 1𝑜
27 1sdom2 7924 . . . . . . . . . . . . . 14 1𝑜 ≺ 2𝑜
28 ensdomtr 7861 . . . . . . . . . . . . . 14 (({𝑥} ≈ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝑥} ≺ 2𝑜)
2926, 27, 28mp2an 703 . . . . . . . . . . . . 13 {𝑥} ≺ 2𝑜
3024, 29syl6eqbr 4520 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2𝑜)
31 sdomnen 7750 . . . . . . . . . . . 12 (({𝑥} ∪ {𝑦}) ≺ 2𝑜 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3230, 31syl 17 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3332necon2ai 2715 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2𝑜𝑥𝑦)
34 disjsn2 4096 . . . . . . . . . 10 (𝑥𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
3533, 34syl 17 . . . . . . . . 9 (({𝑥} ∪ {𝑦}) ≈ 2𝑜 → ({𝑥} ∩ {𝑦}) = ∅)
3635a1i 11 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (({𝑥} ∪ {𝑦}) ≈ 2𝑜 → ({𝑥} ∩ {𝑦}) = ∅))
37 uneq12 3628 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3837breq1d 4491 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 ↔ ({𝑥} ∪ {𝑦}) ≈ 2𝑜))
39 ineq12 3674 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4039eqeq1d 2516 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4136, 38, 403imtr4d 281 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4241ex 448 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4342exlimdv 1814 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4443exlimiv 1811 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4544imp 443 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4619, 20, 45syl2anb 494 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4718, 46impbid 200 1 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1938  wne 2684  cun 3442  cin 3443  c0 3777  {csn 4028   class class class wbr 4481  Oncon0 5530  suc csuc 5532  1𝑜c1o 7320  2𝑜c2o 7321  cen 7718  csdm 7720
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 6728
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-br 4482  df-opab 4542  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 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702  df-om 6839  df-1o 7327  df-2o 7328  df-er 7509  df-en 7722  df-dom 7723  df-sdom 7724
This theorem is referenced by:  pr2nelem  8590  pm110.643  8762  isprm2lem  15112
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