Theorem pm54.43 9690

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 9657), 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 9689. 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 9860 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 8280	. . . . . . 7 ⊢ 1o ∈ V
2	1	ensn1 8761	. . . . . 6 ⊢ {1o} ≈ 1o
3	2	ensymi 8745	. . . . 5 ⊢ 1o ≈ {1o}
4		entr 8747	. . . . 5 ⊢ ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
5	3, 4	mpan2 687	. . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ≈ {1o})
6		1on 8274	. . . . . . . 8 ⊢ 1o ∈ On
7	6	onirri 6358	. . . . . . 7 ⊢ ¬ 1o ∈ 1o
8		disjsn 4644	. . . . . . 7 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
9	7, 8	mpbir 230	. . . . . 6 ⊢ (1o ∩ {1o}) = ∅
10		unen 8790	. . . . . 6 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
11	9, 10	mpanr2 700	. . . . 5 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
12	11	ex 412	. . . 4 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
13	5, 12	sylan2 592	. . 3 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
14		df-2o 8268	. . . . 5 ⊢ 2o = suc 1o
15		df-suc 6257	. . . . 5 ⊢ suc 1o = (1o ∪ {1o})
16	14, 15	eqtri 2766	. . . 4 ⊢ 2o = (1o ∪ {1o})
17	16	breq2i 5078	. . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ 2o ↔ (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
18	13, 17	syl6ibr 251	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ 2o))
19		en1 8765	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20		en1 8765	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21		sneq 4568	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
22	21	uneq2d 4093	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
23		unidm 4082	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
24	22, 23	eqtr3di 2794	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
25		vex 3426	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
26	25	ensn1 8761	. . . . . . . . . . . . . 14 ⊢ {𝑥} ≈ 1o
27		1sdom2 8951	. . . . . . . . . . . . . 14 ⊢ 1o ≺ 2o
28		ensdomtr 8849	. . . . . . . . . . . . . 14 ⊢ (({𝑥} ≈ 1o ∧ 1o ≺ 2o) → {𝑥} ≺ 2o)
29	26, 27, 28	mp2an 688	. . . . . . . . . . . . 13 ⊢ {𝑥} ≺ 2o
30	24, 29	eqbrtrdi 5109	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≺ 2o)
31		sdomnen 8724	. . . . . . . . . . . 12 ⊢ (({𝑥} ∪ {𝑦}) ≺ 2o → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
33	32	necon2ai 2972	. . . . . . . . . 10 ⊢ (({𝑥} ∪ {𝑦}) ≈ 2o → 𝑥 ≠ 𝑦)
34		disjsn2 4645	. . . . . . . . . 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 4088	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∪ 𝐵) = ({𝑥} ∪ {𝑦}))
38	37	breq1d 5080	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
39		ineq12 4138	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∩ 𝐵) = ({𝑥} ∩ {𝑦}))
40	39	eqeq1d 2740	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∩ 𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
41	36, 38, 40	3imtr4d 293	. . . . . . 7 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
42	41	ex 412	. . . . . 6 ⊢ (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
43	42	exlimdv 1937	. . . . 5 ⊢ (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
44	43	exlimiv 1934	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
45	44	imp 406	. . 3 ⊢ ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
46	19, 20, 45	syl2anb 597	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
47	18, 46	impbid 211	1 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942   ∪ cun 3881   ∩ cin 3882  ∅c0 4253  {csn 4558   class class class wbr 5070  suc csuc 6253  1_oc1o 8260  2_oc2o 8261   ≈ cen 8688   ≺ csdm 8690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-2o 8268  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694

This theorem is referenced by:  pr2nelem  9691  dju1p1e2  9860