Theorem pm54.43 7310

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 6519	. . . . . . . 8 ⊢ 1o ∈ On
2	1	elexi 2786	. . . . . . 7 ⊢ 1o ∈ V
3	2	ensn1 6898	. . . . . 6 ⊢ {1o} ≈ 1o
4	3	ensymi 6884	. . . . 5 ⊢ 1o ≈ {1o}
5		entr 6886	. . . . 5 ⊢ ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
6	4, 5	mpan2 425	. . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ≈ {1o})
7	1	onirri 4596	. . . . . . 7 ⊢ ¬ 1o ∈ 1o
8		disjsn 3697	. . . . . . 7 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
9	7, 8	mpbir 146	. . . . . 6 ⊢ (1o ∩ {1o}) = ∅
10		unen 6919	. . . . . 6 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
11	9, 10	mpanr2 438	. . . . 5 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
12	11	ex 115	. . . 4 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
13	6, 12	sylan2 286	. . 3 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
14		df-2o 6513	. . . . 5 ⊢ 2o = suc 1o
15		df-suc 4423	. . . . 5 ⊢ suc 1o = (1o ∪ {1o})
16	14, 15	eqtri 2227	. . . 4 ⊢ 2o = (1o ∪ {1o})
17	16	breq2i 4056	. . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ 2o ↔ (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
18	13, 17	imbitrrdi 162	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ 2o))
19		en1 6901	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20		en1 6901	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21		1nen2 6970	. . . . . . . . . . . . 13 ⊢ ¬ 1o ≈ 2o
22	21	a1i 9	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ¬ 1o ≈ 2o)
23		sneq 3646	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
24	23	uneq2d 3329	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
25		unidm 3318	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
26	24, 25	eqtr3di 2254	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
27		vex 2776	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
28	27	ensn1 6898	. . . . . . . . . . . . . . 15 ⊢ {𝑥} ≈ 1o
29	26, 28	eqbrtrdi 4087	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≈ 1o)
30	29	ensymd 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → 1o ≈ ({𝑥} ∪ {𝑦}))
31		entr 6886	. . . . . . . . . . . . 13 ⊢ ((1o ≈ ({𝑥} ∪ {𝑦}) ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
32	30, 31	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
33	22, 32	mtand 667	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
34	33	necon2ai 2431	. . . . . . . . . 10 ⊢ (({𝑥} ∪ {𝑦}) ≈ 2o → 𝑥 ≠ 𝑦)
35		disjsn2 3698	. . . . . . . . . 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 3324	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∪ 𝐵) = ({𝑥} ∪ {𝑦}))
39	38	breq1d 4058	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
40		ineq12 3371	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∩ 𝐵) = ({𝑥} ∩ {𝑦}))
41	40	eqeq1d 2215	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∩ 𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
42	37, 39, 41	3imtr4d 203	. . . . . . 7 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
43	42	ex 115	. . . . . 6 ⊢ (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
44	43	exlimdv 1843	. . . . 5 ⊢ (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
45	44	exlimiv 1622	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
46	45	imp 124	. . 3 ⊢ ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
47	19, 20, 46	syl2anb 291	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
48	18, 47	impbid 129	1 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177   ≠ wne 2377   ∪ cun 3166   ∩ cin 3167  ∅c0 3462  {csn 3635   class class class wbr 4048  Oncon0 4415  suc csuc 4417  1_oc1o 6505  2_oc2o 6506   ≈ cen 6835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-1o 6512  df-2o 6513  df-er 6630  df-en 6838

This theorem is referenced by:  pr2nelem  7311  dju1p1e2  7318