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Theorem pm54.43 9139
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 9107), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} which is the same as 𝐴 ≈ 1o by pm54.43lem 9138. 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 9314 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 7834 . . . . . . 7 1o ∈ V
21ensn1 8286 . . . . . 6 {1o} ≈ 1o
32ensymi 8272 . . . . 5 1o ≈ {1o}
4 entr 8274 . . . . 5 ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
53, 4mpan2 682 . . . 4 (𝐵 ≈ 1o𝐵 ≈ {1o})
6 1on 7833 . . . . . . . 8 1o ∈ On
76onirri 6069 . . . . . . 7 ¬ 1o ∈ 1o
8 disjsn 4465 . . . . . . 7 ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
97, 8mpbir 223 . . . . . 6 (1o ∩ {1o}) = ∅
10 unen 8309 . . . . . 6 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ ((𝐴𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴𝐵) ≈ (1o ∪ {1o}))
119, 10mpanr2 695 . . . . 5 (((𝐴 ≈ 1o𝐵 ≈ {1o}) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (1o ∪ {1o}))
1211ex 403 . . . 4 ((𝐴 ≈ 1o𝐵 ≈ {1o}) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
135, 12sylan2 586 . . 3 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ (1o ∪ {1o})))
14 df-2o 7827 . . . . 5 2o = suc 1o
15 df-suc 5969 . . . . 5 suc 1o = (1o ∪ {1o})
1614, 15eqtri 2849 . . . 4 2o = (1o ∪ {1o})
1716breq2i 4881 . . 3 ((𝐴𝐵) ≈ 2o ↔ (𝐴𝐵) ≈ (1o ∪ {1o}))
1813, 17syl6ibr 244 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ → (𝐴𝐵) ≈ 2o))
19 en1 8289 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20 en1 8289 . . 3 (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21 unidm 3983 . . . . . . . . . . . . . 14 ({𝑥} ∪ {𝑥}) = {𝑥}
22 sneq 4407 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2322uneq2d 3994 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
2421, 23syl5reqr 2876 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25 vex 3417 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2625ensn1 8286 . . . . . . . . . . . . . 14 {𝑥} ≈ 1o
27 1sdom2 8428 . . . . . . . . . . . . . 14 1o ≺ 2o
28 ensdomtr 8365 . . . . . . . . . . . . . 14 (({𝑥} ≈ 1o ∧ 1o ≺ 2o) → {𝑥} ≺ 2o)
2926, 27, 28mp2an 683 . . . . . . . . . . . . 13 {𝑥} ≺ 2o
3024, 29syl6eqbr 4912 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2o)
31 sdomnen 8251 . . . . . . . . . . . 12 (({𝑥} ∪ {𝑦}) ≺ 2o → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
3230, 31syl 17 . . . . . . . . . . 11 (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
3332necon2ai 3028 . . . . . . . . . 10 (({𝑥} ∪ {𝑦}) ≈ 2o𝑥𝑦)
34 disjsn2 4466 . . . . . . . . . 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 3989 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∪ {𝑦}))
3837breq1d 4883 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
39 ineq12 4036 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴𝐵) = ({𝑥} ∩ {𝑦}))
4039eqeq1d 2827 . . . . . . . 8 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
4136, 38, 403imtr4d 286 . . . . . . 7 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4241ex 403 . . . . . 6 (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4342exlimdv 2032 . . . . 5 (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4443exlimiv 2029 . . . 4 (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅)))
4544imp 397 . . 3 ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4619, 20, 45syl2anb 591 . 2 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) ≈ 2o → (𝐴𝐵) = ∅))
4718, 46impbid 204 1 ((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2o))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1656  wex 1878  wcel 2164  wne 2999  cun 3796  cin 3797  c0 4144  {csn 4397   class class class wbr 4873  suc csuc 5965  1oc1o 7819  2oc2o 7820  cen 8219  csdm 8221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-om 7327  df-1o 7826  df-2o 7827  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225
This theorem is referenced by:  pr2nelem  9140  pm110.643  9314
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