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 9027
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 8995), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} which is the same as 𝐴 ≈ 1𝑜 by pm54.43lem 9026. 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 9202 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 1oex 7722 . . . . . . 7 1𝑜 ∈ V
21ensn1 8174 . . . . . 6 {1𝑜} ≈ 1𝑜
32ensymi 8160 . . . . 5 1𝑜 ≈ {1𝑜}
4 entr 8162 . . . . 5 ((𝐵 ≈ 1𝑜 ∧ 1𝑜 ≈ {1𝑜}) → 𝐵 ≈ {1𝑜})
53, 4mpan2 665 . . . 4 (𝐵 ≈ 1𝑜𝐵 ≈ {1𝑜})
6 1on 7721 . . . . . . . 8 1𝑜 ∈ On
76onirri 5978 . . . . . . 7 ¬ 1𝑜 ∈ 1𝑜
8 disjsn 4384 . . . . . . 7 ((1𝑜 ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ 1𝑜)
97, 8mpbir 221 . . . . . 6 (1𝑜 ∩ {1𝑜}) = ∅
10 unen 8197 . . . . . 6 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ ((𝐴𝐵) = ∅ ∧ (1𝑜 ∩ {1𝑜}) = ∅)) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
119, 10mpanr2 678 . . . . 5 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1211ex 397 . . . 4 ((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
135, 12sylan2 574 . . 3 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
14 df-2o 7715 . . . . 5 2𝑜 = suc 1𝑜
15 df-suc 5873 . . . . 5 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
1614, 15eqtri 2793 . . . 4 2𝑜 = (1𝑜 ∪ {1𝑜})
1716breq2i 4795 . . 3 ((𝐴𝐵) ≈ 2𝑜 ↔ (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1813, 17syl6ibr 242 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2𝑜))
19 en1 8177 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 8177 . . 3 (𝐵 ≈ 1𝑜 ↔ ∃𝑦 𝐵 = {𝑦})
21 unidm 3908 . . . . . . . . . . . . . 14 ({𝑥} ∪ {𝑥}) = {𝑥}
22 sneq 4327 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2322uneq2d 3919 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
2421, 23syl5reqr 2820 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25 vex 3354 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2625ensn1 8174 . . . . . . . . . . . . . 14 {𝑥} ≈ 1𝑜
27 1sdom2 8316 . . . . . . . . . . . . . 14 1𝑜 ≺ 2𝑜
28 ensdomtr 8253 . . . . . . . . . . . . . 14 (({𝑥} ≈ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝑥} ≺ 2𝑜)
2926, 27, 28mp2an 666 . . . . . . . . . . . . 13 {𝑥} ≺ 2𝑜
3024, 29syl6eqbr 4826 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2𝑜)
31 sdomnen 8139 . . . . . . . . . . . 12 (({𝑥} ∪ {𝑦}) ≺ 2𝑜 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3230, 31syl 17 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3332necon2ai 2972 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2𝑜𝑥𝑦)
34 disjsn2 4385 . . . . . . . . . 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 3914 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3837breq1d 4797 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 ↔ ({𝑥} ∪ {𝑦}) ≈ 2𝑜))
39 ineq12 3961 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4039eqeq1d 2773 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4136, 38, 403imtr4d 283 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4241ex 397 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4342exlimdv 2013 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4443exlimiv 2010 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4544imp 393 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4619, 20, 45syl2anb 579 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4718, 46impbid 202 1 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  cun 3722  cin 3723  c0 4064  {csn 4317   class class class wbr 4787  suc csuc 5869  1𝑜c1o 7707  2𝑜c2o 7708  cen 8107  csdm 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-om 7214  df-1o 7714  df-2o 7715  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113
This theorem is referenced by:  pr2nelem  9028  pm110.643  9202
  Copyright terms: Public domain W3C validator