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Theorem pm54.43 8771
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 8739), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} which is the same as 𝐴 ≈ 1𝑜 by pm54.43lem 8770. 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 8944 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 7513 . . . . . . . 8 1𝑜 ∈ On
21elexi 3204 . . . . . . 7 1𝑜 ∈ V
32ensn1 7965 . . . . . 6 {1𝑜} ≈ 1𝑜
43ensymi 7951 . . . . 5 1𝑜 ≈ {1𝑜}
5 entr 7953 . . . . 5 ((𝐵 ≈ 1𝑜 ∧ 1𝑜 ≈ {1𝑜}) → 𝐵 ≈ {1𝑜})
64, 5mpan2 706 . . . 4 (𝐵 ≈ 1𝑜𝐵 ≈ {1𝑜})
71onirri 5796 . . . . . . 7 ¬ 1𝑜 ∈ 1𝑜
8 disjsn 4221 . . . . . . 7 ((1𝑜 ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ 1𝑜)
97, 8mpbir 221 . . . . . 6 (1𝑜 ∩ {1𝑜}) = ∅
10 unen 7985 . . . . . 6 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ ((𝐴𝐵) = ∅ ∧ (1𝑜 ∩ {1𝑜}) = ∅)) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
119, 10mpanr2 719 . . . . 5 (((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1211ex 450 . . . 4 ((𝐴 ≈ 1𝑜𝐵 ≈ {1𝑜}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
136, 12sylan2 491 . . 3 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜})))
14 df-2o 7507 . . . . 5 2𝑜 = suc 1𝑜
15 df-suc 5691 . . . . 5 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
1614, 15eqtri 2648 . . . 4 2𝑜 = (1𝑜 ∪ {1𝑜})
1716breq2i 4626 . . 3 ((𝐴𝐵) ≈ 2𝑜 ↔ (𝐴𝐵) ≈ (1𝑜 ∪ {1𝑜}))
1813, 17syl6ibr 242 . 2 ((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2𝑜))
19 en1 7968 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 7968 . . 3 (𝐵 ≈ 1𝑜 ↔ ∃𝑦 𝐵 = {𝑦})
21 unidm 3739 . . . . . . . . . . . . . 14 ({𝑥} ∪ {𝑥}) = {𝑥}
22 sneq 4163 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2322uneq2d 3750 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
2421, 23syl5reqr 2675 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25 vex 3194 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2625ensn1 7965 . . . . . . . . . . . . . 14 {𝑥} ≈ 1𝑜
27 1sdom2 8104 . . . . . . . . . . . . . 14 1𝑜 ≺ 2𝑜
28 ensdomtr 8041 . . . . . . . . . . . . . 14 (({𝑥} ≈ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝑥} ≺ 2𝑜)
2926, 27, 28mp2an 707 . . . . . . . . . . . . 13 {𝑥} ≺ 2𝑜
3024, 29syl6eqbr 4657 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2𝑜)
31 sdomnen 7929 . . . . . . . . . . . 12 (({𝑥} ∪ {𝑦}) ≺ 2𝑜 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3230, 31syl 17 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
3332necon2ai 2825 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2𝑜𝑥𝑦)
34 disjsn2 4222 . . . . . . . . . 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 3745 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3837breq1d 4628 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 ↔ ({𝑥} ∪ {𝑦}) ≈ 2𝑜))
39 ineq12 3792 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4039eqeq1d 2628 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4136, 38, 403imtr4d 283 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4241ex 450 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4342exlimdv 1863 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4443exlimiv 1860 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅)))
4544imp 445 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2𝑜 → (𝐴𝐵) = ∅))
4619, 20, 45syl2anb 496 . 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 384   = wceq 1480  wex 1701  wcel 1992  wne 2796  cun 3558  cin 3559  c0 3896  {csn 4153   class class class wbr 4618  Oncon0 5685  suc csuc 5687  1𝑜c1o 7499  2𝑜c2o 7500  cen 7897  csdm 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-2o 7507  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903
This theorem is referenced by:  pr2nelem  8772  pm110.643  8944  isprm2lem  15313
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