Theorem pm54.43 9423

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 9391), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1_o} which is the same as 𝐴 ≈ 1_o by pm54.43lem 9422. 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 9593 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 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))

Proof of Theorem pm54.43

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 8104	. . . . . . 7 ⊢ 1o ∈ V
2	1	ensn1 8567	. . . . . 6 ⊢ {1o} ≈ 1o
3	2	ensymi 8553	. . . . 5 ⊢ 1o ≈ {1o}
4		entr 8555	. . . . 5 ⊢ ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
5	3, 4	mpan2 689	. . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ≈ {1o})
6		1on 8103	. . . . . . . 8 ⊢ 1o ∈ On
7	6	onirri 6291	. . . . . . 7 ⊢ ¬ 1o ∈ 1o
8		disjsn 4640	. . . . . . 7 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
9	7, 8	mpbir 233	. . . . . 6 ⊢ (1o ∩ {1o}) = ∅
10		unen 8590	. . . . . 6 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
11	9, 10	mpanr2 702	. . . . 5 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
12	11	ex 415	. . . 4 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
13	5, 12	sylan2 594	. . 3 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
14		df-2o 8097	. . . . 5 ⊢ 2o = suc 1o
15		df-suc 6191	. . . . 5 ⊢ suc 1o = (1o ∪ {1o})
16	14, 15	eqtri 2844	. . . 4 ⊢ 2o = (1o ∪ {1o})
17	16	breq2i 5066	. . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ 2o ↔ (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
18	13, 17	syl6ibr 254	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ 2o))
19		en1 8570	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20		en1 8570	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21		unidm 4127	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
22		sneq 4570	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
23	22	uneq2d 4138	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
24	21, 23	syl5reqr 2871	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25		vex 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
26	25	ensn1 8567	. . . . . . . . . . . . . 14 ⊢ {𝑥} ≈ 1o
27		1sdom2 8711	. . . . . . . . . . . . . 14 ⊢ 1o ≺ 2o
28		ensdomtr 8647	. . . . . . . . . . . . . 14 ⊢ (({𝑥} ≈ 1o ∧ 1o ≺ 2o) → {𝑥} ≺ 2o)
29	26, 27, 28	mp2an 690	. . . . . . . . . . . . 13 ⊢ {𝑥} ≺ 2o
30	24, 29	eqbrtrdi 5097	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2o)
31		sdomnen 8532	. . . . . . . . . . . 12 ⊢ (({𝑥} ∪ {𝑦}) ≺ 2o → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
33	32	necon2ai 3045	. . . . . . . . . 10 ⊢ (({𝑥} ∪ {𝑦}) ≈ 2o → 𝑥 ≠ 𝑦)
34		disjsn2 4641	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (({𝑥} ∪ {𝑦}) ≈ 2o → ({𝑥} ∩ {𝑦}) = ∅)
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (({𝑥} ∪ {𝑦}) ≈ 2o → ({𝑥} ∩ {𝑦}) = ∅))
37		uneq12 4133	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∪ 𝐵) = ({𝑥} ∪ {𝑦}))
38	37	breq1d 5068	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
39		ineq12 4183	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∩ 𝐵) = ({𝑥} ∩ {𝑦}))
40	39	eqeq1d 2823	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∩ 𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
41	36, 38, 40	3imtr4d 296	. . . . . . 7 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
42	41	ex 415	. . . . . 6 ⊢ (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
43	42	exlimdv 1930	. . . . 5 ⊢ (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
44	43	exlimiv 1927	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
45	44	imp 409	. . 3 ⊢ ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
46	19, 20, 45	syl2anb 599	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
47	18, 46	impbid 214	1 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016   ∪ cun 3933   ∩ cin 3934  ∅c0 4290  {csn 4560   class class class wbr 5058  suc csuc 6187  1_oc1o 8089  2_oc2o 8090   ≈ cen 8500   ≺ csdm 8502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7575  df-1o 8096  df-2o 8097  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506

This theorem is referenced by:  pr2nelem  9424  dju1p1e2  9593