Theorem pm54.43 4552

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 4813), so that their A ∈ 1 means, in our notation, A ∈ {x∣(card ‘x) = 1_o} i.e. (card ‘A) = 1_o (by elab 1893) i.e. A ≈ 1_o (by carden 4811 and cardnn 4804). 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 4903 shows the derivation of 1+1=2 for cardinal numbers from this theorem.

Assertion

Ref	Expression
pm54.43	⊢ ((A ≈ 1o ⋀ B ≈ 1o) → ((A ∩ B) = ∅ ↔ (A ∪ B) ≈ 2o))

Proof of Theorem pm54.43

Step	Hyp	Ref	Expression
1		1on 4128	. . . . . . . 8 ⊢ 1o ∈ On
2	1	onirr 3092	. . . . . . 7 ⊢ ¬ 1o ∈ 1o
3		disjsn 2437	. . . . . . 7 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
4	2, 3	mpbir 190	. . . . . 6 ⊢ (1o ∩ {1o}) = ∅
5		unen 4420	. . . . . 6 ⊢ (((A ≈ 1o ⋀ B ≈ {1o}) ⋀ ((A ∩ B) = ∅ ⋀ (1o ∩ {1o}) = ∅)) → (A ∪ B) ≈ (1o ∪ {1o}))
6	4, 5	mpanr2 709	. . . . 5 ⊢ (((A ≈ 1o ⋀ B ≈ {1o}) ⋀ (A ∩ B) = ∅) → (A ∪ B) ≈ (1o ∪ {1o}))
7	6	ex 373	. . . 4 ⊢ ((A ≈ 1o ⋀ B ≈ {1o}) → ((A ∩ B) = ∅ → (A ∪ B) ≈ (1o ∪ {1o})))
8	1	elisseti 1814	. . . . . 6 ⊢ 1o ∈ V
9	8	ensn1 4411	. . . . . 6 ⊢ {1o} ≈ 1o
10	8, 9	ensymi 4400	. . . . 5 ⊢ 1o ≈ {1o}
11		entrt 4401	. . . . 5 ⊢ ((B ≈ 1o ⋀ 1o ≈ {1o}) → B ≈ {1o})
12	10, 11	mpan2 695	. . . 4 ⊢ (B ≈ 1o → B ≈ {1o})
13	7, 12	sylan2 451	. . 3 ⊢ ((A ≈ 1o ⋀ B ≈ 1o) → ((A ∩ B) = ∅ → (A ∪ B) ≈ (1o ∪ {1o})))
14		df-2o 4124	. . . . 5 ⊢ 2o = suc 1o
15		df-suc 2949	. . . . 5 ⊢ suc 1o = (1o ∪ {1o})
16	14, 15	eqtr 1492	. . . 4 ⊢ 2o = (1o ∪ {1o})
17	16	breq2i 2622	. . 3 ⊢ ((A ∪ B) ≈ 2o ↔ (A ∪ B) ≈ (1o ∪ {1o}))
18	13, 17	syl6ibr 213	. 2 ⊢ ((A ≈ 1o ⋀ B ≈ 1o) → ((A ∩ B) = ∅ → (A ∪ B) ≈ 2o))
19		sneq 2413	. . . . . . . . . . . . . . 15 ⊢ (x = y → {x} = {y})
20	19	uneq2d 2180	. . . . . . . . . . . . . 14 ⊢ (x = y → ({x} ∪ {x}) = ({x} ∪ {y}))
21		unidm 2171	. . . . . . . . . . . . . 14 ⊢ ({x} ∪ {x}) = {x}
22	20, 21	syl5reqr 1519	. . . . . . . . . . . . 13 ⊢ (x = y → ({x} ∪ {y}) = {x})
23		visset 1809	. . . . . . . . . . . . . . 15 ⊢ x ∈ V
24	23	ensn1 4411	. . . . . . . . . . . . . 14 ⊢ {x} ≈ 1o
25		1sdom2 4511	. . . . . . . . . . . . . 14 ⊢ 1o ≺ 2o
26		ensdomtr 4457	. . . . . . . . . . . . . 14 ⊢ (({x} ≈ 1o ⋀ 1o ≺ 2o) → {x} ≺ 2o)
27	24, 25, 26	mp2an 696	. . . . . . . . . . . . 13 ⊢ {x} ≺ 2o
28	22, 27	syl6eqbr 2647	. . . . . . . . . . . 12 ⊢ (x = y → ({x} ∪ {y}) ≺ 2o)
29		sdomnen 4374	. . . . . . . . . . . 12 ⊢ (({x} ∪ {y}) ≺ 2o → ¬ ({x} ∪ {y}) ≈ 2o)
30	28, 29	syl 10	. . . . . . . . . . 11 ⊢ (x = y → ¬ ({x} ∪ {y}) ≈ 2o)
31	30	necon2ai 1608	. . . . . . . . . 10 ⊢ (({x} ∪ {y}) ≈ 2o → x ≠ y)
32		disjsn2 2438	. . . . . . . . . 10 ⊢ (x ≠ y → ({x} ∩ {y}) = ∅)
33	31, 32	syl 10	. . . . . . . . 9 ⊢ (({x} ∪ {y}) ≈ 2o → ({x} ∩ {y}) = ∅)
34	33	a1i 8	. . . . . . . 8 ⊢ ((A = {x} ⋀ B = {y}) → (({x} ∪ {y}) ≈ 2o → ({x} ∩ {y}) = ∅))
35		uneq12 2175	. . . . . . . . 9 ⊢ ((A = {x} ⋀ B = {y}) → (A ∪ B) = ({x} ∪ {y}))
36	35	breq1d 2624	. . . . . . . 8 ⊢ ((A = {x} ⋀ B = {y}) → ((A ∪ B) ≈ 2o ↔ ({x} ∪ {y}) ≈ 2o))
37		ineq12 2208	. . . . . . . . 9 ⊢ ((A = {x} ⋀ B = {y}) → (A ∩ B) = ({x} ∩ {y}))
38	37	eqeq1d 1480	. . . . . . . 8 ⊢ ((A = {x} ⋀ B = {y}) → ((A ∩ B) = ∅ ↔ ({x} ∩ {y}) = ∅))
39	34, 36, 38	3imtr4d 542	. . . . . . 7 ⊢ ((A = {x} ⋀ B = {y}) → ((A ∪ B) ≈ 2o → (A ∩ B) = ∅))
40	39	ex 373	. . . . . 6 ⊢ (A = {x} → (B = {y} → ((A ∪ B) ≈ 2o → (A ∩ B) = ∅)))
41	40	19.23adv 1212	. . . . 5 ⊢ (A = {x} → (∃y B = {y} → ((A ∪ B) ≈ 2o → (A ∩ B) = ∅)))
42	41	19.23aiv 1293	. . . 4 ⊢ (∃x A = {x} → (∃y B = {y} → ((A ∪ B) ≈ 2o → (A ∩ B) = ∅)))
43	42	imp 350	. . 3 ⊢ ((∃x A = {x} ⋀ ∃y B = {y}) → ((A ∪ B) ≈ 2o → (A ∩ B) = ∅))
44		en1 4413	. . 3 ⊢ (A ≈ 1o ↔ ∃x A = {x})
45		en1 4413	. . 3 ⊢ (B ≈ 1o ↔ ∃y B = {y})
46	43, 44, 45	syl2anb 455	. 2 ⊢ ((A ≈ 1o ⋀ B ≈ 1o) → ((A ∪ B) ≈ 2o → (A ∩ B) = ∅))
47	18, 46	impbid 515	1 ⊢ ((A ≈ 1o ⋀ B ≈ 1o) → ((A ∩ B) = ∅ ↔ (A ∪ B) ≈ 2o))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∃wex 978   ≠ wne 1582   ∪ cun 2041   ∩ cin 2042  ∅c0 2276  {csn 2405   class class class wbr 2614  Oncon0 2943  suc csuc 2945  1_oc1o 4118  2_oc2o 4119   ≈ cen 4354   ≺ csdm 4356

This theorem is referenced by:  pm110.643 4903

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-1o 4123  df-2o 4124  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359

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