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Theorem pm54.43 10016
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 9983), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} which is the same as 𝐴 ≈ 1o by pm54.43lem 10015. 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 dju1p1e2 10188 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 8490 . . . . . . 7 1o ∈ V
21ensn1 9033 . . . . . 6 {1o} ≈ 1o
32ensymi 9016 . . . . 5 1o ≈ {1o}
4 entr 9018 . . . . 5 ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
53, 4mpan2 690 . . . 4 (𝐵 ≈ 1o𝐵 ≈ {1o})
6 1on 8492 . . . . . . . 8 1o ∈ On
76onirri 6476 . . . . . . 7 ¬ 1o ∈ 1o
8 disjsn 4711 . . . . . . 7 ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
97, 8mpbir 230 . . . . . 6 (1o ∩ {1o}) = ∅
10 unen 9062 . . . . . 6 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ ((𝐴𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴𝐵) ≈ (1o ∪ {1o}))
119, 10mpanr2 703 . . . . 5 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1o ∪ {1o}))
1211ex 412 . . . 4 ((𝐴 ≈ 1o𝐵 ≈ {1o}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
135, 12sylan2 592 . . 3 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
14 df-2o 8481 . . . . 5 2o = suc 1o
15 df-suc 6369 . . . . 5 suc 1o = (1o ∪ {1o})
1614, 15eqtri 2755 . . . 4 2o = (1o ∪ {1o})
1716breq2i 5150 . . 3 ((𝐴𝐵) ≈ 2o ↔ (𝐴𝐵) ≈ (1o ∪ {1o}))
1813, 17imbitrrdi 251 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2o))
19 en1 9037 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 9037 . . 3 (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21 sneq 4634 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2221uneq2d 4159 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
23 unidm 4148 . . . . . . . . . . . . . 14 ({𝑥} ∪ {𝑥}) = {𝑥}
2422, 23eqtr3di 2782 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25 vex 3473 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2625ensn1 9033 . . . . . . . . . . . . . 14 {𝑥} ≈ 1o
27 1sdom2 9256 . . . . . . . . . . . . . 14 1o ≺ 2o
28 ensdomtr 9129 . . . . . . . . . . . . . 14 (({𝑥} ≈ 1o ∧ 1o ≺ 2o) → {𝑥} ≺ 2o)
2926, 27, 28mp2an 691 . . . . . . . . . . . . 13 {𝑥} ≺ 2o
3024, 29eqbrtrdi 5181 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2o)
31 sdomnen 8993 . . . . . . . . . . . 12 (({𝑥} ∪ {𝑦}) ≺ 2o → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
3230, 31syl 17 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
3332necon2ai 2965 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2o𝑥𝑦)
34 disjsn2 4712 . . . . . . . . . 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 4154 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3837breq1d 5152 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
39 ineq12 4203 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4039eqeq1d 2729 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4136, 38, 403imtr4d 294 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4241ex 412 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4342exlimdv 1929 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4443exlimiv 1926 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4544imp 406 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4619, 20, 45syl2anb 597 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4718, 46impbid 211 1 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2o))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1534  wex 1774  wcel 2099  wne 2935  cun 3942  cin 3943  c0 4318  {csn 4624   class class class wbr 5142  suc csuc 6365  1oc1o 8473  2oc2o 8474  cen 8952  csdm 8954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1o 8480  df-2o 8481  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958
This theorem is referenced by:  pr2nelemOLD  10018  dju1p1e2  10188
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