MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm54.43 Structured version   Visualization version   GIF version

Theorem pm54.43 8519
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 8487), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} which is the same as 𝐴 ≈ 1𝑜 by pm54.43lem 8518. 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 8692 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 7266 . . . . . . . 8 1𝑜 ∈ On
21elexi 3076 . . . . . . 7 1𝑜 ∈ V
32ensn1 7717 . . . . . 6 {1𝑜} ≈ 1𝑜
43ensymi 7703 . . . . 5 1𝑜 ≈ {1𝑜}
5 entr 7705 . . . . 5 ((𝐵 ≈ 1𝑜 ∧ 1𝑜 ≈ {1𝑜}) → 𝐵 ≈ {1𝑜})
64, 5mpan2 694 . . . 4 (𝐵 ≈ 1𝑜𝐵 ≈ {1𝑜})
71onirri 5580 . . . . . . 7 ¬ 1𝑜 ∈ 1𝑜
8 disjsn 4060 . . . . . . 7 ((1𝑜 ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ 1𝑜)
97, 8mpbir 216 . . . . . 6 (1𝑜 ∩ {1𝑜}) = ∅
10 unen 7736 . . . . . 6 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ ((𝐴𝐵) = ∅ ∧ (1𝑜 ∩ {1𝑜}) = ∅)) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
119, 10mpanr2 707 . . . . 5 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1211ex 443 . . . 4 ((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
136, 12sylan2 484 . . 3 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
14 df-2o 7260 . . . . 5 2𝑜 = suc 1𝑜
15 df-suc 5480 . . . . 5 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
1614, 15eqtri 2527 . . . 4 2𝑜 = (1𝑜 ∪ {1𝑜})
1716breq2i 4442 . . 3 ((𝐴𝐵) ≈ 2𝑜 ↔ (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1813, 17syl6ibr 237 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2𝑜))
19 en1 7720 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 7720 . . 3 (𝐵 ≈ 1𝑜 ↔ ∃𝑦 𝐵 = {𝑦})
21 unidm 3604 . . . . . . . . . . . . . 14 ({𝑥} ∪ {𝑥}) = {𝑥}
22 sneq 4005 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2322uneq2d 3615 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
2421, 23syl5reqr 2554 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25 vex 3069 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2625ensn1 7717 . . . . . . . . . . . . . 14 {𝑥} ≈ 1𝑜
27 1sdom2 7855 . . . . . . . . . . . . . 14 1𝑜 ≺ 2𝑜
28 ensdomtr 7792 . . . . . . . . . . . . . 14 (({𝑥} ≈ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝑥} ≺ 2𝑜)
2926, 27, 28mp2an 695 . . . . . . . . . . . . 13 {𝑥} ≺ 2𝑜
3024, 29syl6eqbr 4472 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2𝑜)
31 sdomnen 7681 . . . . . . . . . . . 12 (({𝑥} ∪ {𝑦}) ≺ 2𝑜 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3230, 31syl 17 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3332necon2ai 2706 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2𝑜𝑥𝑦)
34 disjsn2 4061 . . . . . . . . . 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 3610 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3837breq1d 4444 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 ↔ ({𝑥} ∪ {𝑦}) ≈ 2𝑜))
39 ineq12 3656 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4039eqeq1d 2507 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4136, 38, 403imtr4d 278 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4241ex 443 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4342exlimdv 1810 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4443exlimiv 1807 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4544imp 438 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4619, 20, 45syl2anb 489 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4718, 46impbid 197 1 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 191  wa 378   = wceq 1468  wex 1692  wcel 1937  wne 2675  cun 3424  cin 3425  c0 3757  {csn 3995   class class class wbr 4434  Oncon0 5474  suc csuc 5476  1𝑜c1o 7252  2𝑜c2o 7253  cen 7649  csdm 7651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-br 4435  df-opab 4494  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-om 6770  df-1o 7259  df-2o 7260  df-er 7440  df-en 7653  df-dom 7654  df-sdom 7655
This theorem is referenced by:  pr2nelem  8520  pm110.643  8692  isprm2lem  14793
  Copyright terms: Public domain W3C validator