Theorem pm54.43 6459

Description: Theorem *54.43 of [WhiteheadRussell] p. 360. (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		1on 6031	. . . . . . . 8 ⊢ 1𝑜 ∈ On
2	1	elexi 2611	. . . . . . 7 ⊢ 1𝑜 ∈ V
3	2	ensn1 6299	. . . . . 6 ⊢ {1𝑜} ≈ 1𝑜
4	3	ensymi 6285	. . . . 5 ⊢ 1𝑜 ≈ {1𝑜}
5		entr 6287	. . . . 5 ⊢ ((𝐵 ≈ 1𝑜 ∧ 1𝑜 ≈ {1𝑜}) → 𝐵 ≈ {1𝑜})
6	4, 5	mpan2 415	. . . 4 ⊢ (𝐵 ≈ 1𝑜 → 𝐵 ≈ {1𝑜})
7	1	onirri 4286	. . . . . . 7 ⊢ ¬ 1𝑜 ∈ 1𝑜
8		disjsn 3454	. . . . . . 7 ⊢ ((1𝑜 ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ 1𝑜)
9	7, 8	mpbir 144	. . . . . 6 ⊢ (1𝑜 ∩ {1𝑜}) = ∅
10		unen 6316	. . . . . 6 ⊢ (((𝐴 ≈ 1𝑜 ∧ 𝐵 ≈ {1𝑜}) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (1𝑜 ∩ {1𝑜}) = ∅)) → (𝐴 ∪ 𝐵) ≈ (1𝑜 ∪ {1𝑜}))
11	9, 10	mpanr2 428	. . . . 5 ⊢ (((𝐴 ≈ 1𝑜 ∧ 𝐵 ≈ {1𝑜}) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ (1𝑜 ∪ {1𝑜}))
12	11	ex 113	. . . 4 ⊢ ((𝐴 ≈ 1𝑜 ∧ 𝐵 ≈ {1𝑜}) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1𝑜 ∪ {1𝑜})))
13	6, 12	sylan2 280	. . 3 ⊢ ((𝐴 ≈ 1𝑜 ∧ 𝐵 ≈ 1𝑜) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1𝑜 ∪ {1𝑜})))
14		df-2o 6025	. . . . 5 ⊢ 2𝑜 = suc 1𝑜
15		df-suc 4126	. . . . 5 ⊢ suc 1𝑜 = (1𝑜 ∪ {1𝑜})
16	14, 15	eqtri 2101	. . . 4 ⊢ 2𝑜 = (1𝑜 ∪ {1𝑜})
17	16	breq2i 3793	. . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ 2𝑜 ↔ (𝐴 ∪ 𝐵) ≈ (1𝑜 ∪ {1𝑜}))
18	13, 17	syl6ibr 160	. 2 ⊢ ((𝐴 ≈ 1𝑜 ∧ 𝐵 ≈ 1𝑜) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ 2𝑜))
19		en1 6302	. . 3 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
20		en1 6302	. . 3 ⊢ (𝐵 ≈ 1𝑜 ↔ ∃𝑦 𝐵 = {𝑦})
21		1nen2 6347	. . . . . . . . . . . . 13 ⊢ ¬ 1𝑜 ≈ 2𝑜
22	21	a1i 9	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ¬ 1𝑜 ≈ 2𝑜)
23		unidm 3115	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
24		sneq 3409	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
25	24	uneq2d 3126	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
26	23, 25	syl5reqr 2128	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
27		vex 2604	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
28	27	ensn1 6299	. . . . . . . . . . . . . . 15 ⊢ {𝑥} ≈ 1𝑜
29	26, 28	syl6eqbr 3822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≈ 1𝑜)
30	29	ensymd 6286	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → 1𝑜 ≈ ({𝑥} ∪ {𝑦}))
31		entr 6287	. . . . . . . . . . . . 13 ⊢ ((1𝑜 ≈ ({𝑥} ∪ {𝑦}) ∧ ({𝑥} ∪ {𝑦}) ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
32	30, 31	sylan 277	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ ({𝑥} ∪ {𝑦}) ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
33	22, 32	mtand 623	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2𝑜)
34	33	necon2ai 2299	. . . . . . . . . 10 ⊢ (({𝑥} ∪ {𝑦}) ≈ 2𝑜 → 𝑥 ≠ 𝑦)
35		disjsn2 3455	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
36	34, 35	syl 14	. . . . . . . . 9 ⊢ (({𝑥} ∪ {𝑦}) ≈ 2𝑜 → ({𝑥} ∩ {𝑦}) = ∅)
37	36	a1i 9	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (({𝑥} ∪ {𝑦}) ≈ 2𝑜 → ({𝑥} ∩ {𝑦}) = ∅))
38		uneq12 3121	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∪ 𝐵) = ({𝑥} ∪ {𝑦}))
39	38	breq1d 3795	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2𝑜 ↔ ({𝑥} ∪ {𝑦}) ≈ 2𝑜))
40		ineq12 3162	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∩ 𝐵) = ({𝑥} ∩ {𝑦}))
41	40	eqeq1d 2089	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∩ 𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
42	37, 39, 41	3imtr4d 201	. . . . . . 7 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2𝑜 → (𝐴 ∩ 𝐵) = ∅))
43	42	ex 113	. . . . . 6 ⊢ (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2𝑜 → (𝐴 ∩ 𝐵) = ∅)))
44	43	exlimdv 1740	. . . . 5 ⊢ (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2𝑜 → (𝐴 ∩ 𝐵) = ∅)))
45	44	exlimiv 1529	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2𝑜 → (𝐴 ∩ 𝐵) = ∅)))
46	45	imp 122	. . 3 ⊢ ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2𝑜 → (𝐴 ∩ 𝐵) = ∅))
47	19, 20, 46	syl2anb 285	. 2 ⊢ ((𝐴 ≈ 1𝑜 ∧ 𝐵 ≈ 1𝑜) → ((𝐴 ∪ 𝐵) ≈ 2𝑜 → (𝐴 ∩ 𝐵) = ∅))
48	18, 47	impbid 127	1 ⊢ ((𝐴 ≈ 1𝑜 ∧ 𝐵 ≈ 1𝑜) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2𝑜))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433   ≠ wne 2245   ∪ cun 2971   ∩ cin 2972  ∅c0 3251  {csn 3398   class class class wbr 3785  Oncon0 4118  suc csuc 4120  1_𝑜c1o 6017  2_𝑜c2o 6018   ≈ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-1o 6024  df-2o 6025  df-er 6129  df-en 6245

This theorem is referenced by:  pr2nelem  6460