Theorem pm54.43 7039

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 6313	. . . . . . . 8 ⊢ 1o ∈ On
2	1	elexi 2693	. . . . . . 7 ⊢ 1o ∈ V
3	2	ensn1 6683	. . . . . 6 ⊢ {1o} ≈ 1o
4	3	ensymi 6669	. . . . 5 ⊢ 1o ≈ {1o}
5		entr 6671	. . . . 5 ⊢ ((𝐵 ≈ 1o ∧ 1o ≈ {1o}) → 𝐵 ≈ {1o})
6	4, 5	mpan2 421	. . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ≈ {1o})
7	1	onirri 4453	. . . . . . 7 ⊢ ¬ 1o ∈ 1o
8		disjsn 3580	. . . . . . 7 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
9	7, 8	mpbir 145	. . . . . 6 ⊢ (1o ∩ {1o}) = ∅
10		unen 6703	. . . . . 6 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (1o ∩ {1o}) = ∅)) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
11	9, 10	mpanr2 434	. . . . 5 ⊢ (((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
12	11	ex 114	. . . 4 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ {1o}) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
13	6, 12	sylan2 284	. . 3 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o})))
14		df-2o 6307	. . . . 5 ⊢ 2o = suc 1o
15		df-suc 4288	. . . . 5 ⊢ suc 1o = (1o ∪ {1o})
16	14, 15	eqtri 2158	. . . 4 ⊢ 2o = (1o ∪ {1o})
17	16	breq2i 3932	. . 3 ⊢ ((𝐴 ∪ 𝐵) ≈ 2o ↔ (𝐴 ∪ 𝐵) ≈ (1o ∪ {1o}))
18	13, 17	syl6ibr 161	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∪ 𝐵) ≈ 2o))
19		en1 6686	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
20		en1 6686	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑦 𝐵 = {𝑦})
21		1nen2 6748	. . . . . . . . . . . . 13 ⊢ ¬ 1o ≈ 2o
22	21	a1i 9	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ¬ 1o ≈ 2o)
23		unidm 3214	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥} ∪ {𝑥}) = {𝑥}
24		sneq 3533	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
25	24	uneq2d 3225	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑥}) = ({𝑥} ∪ {𝑦}))
26	23, 25	syl5reqr 2185	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) = {𝑥})
27		vex 2684	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
28	27	ensn1 6683	. . . . . . . . . . . . . . 15 ⊢ {𝑥} ≈ 1o
29	26, 28	eqbrtrdi 3962	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ {𝑦}) ≈ 1o)
30	29	ensymd 6670	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → 1o ≈ ({𝑥} ∪ {𝑦}))
31		entr 6671	. . . . . . . . . . . . 13 ⊢ ((1o ≈ ({𝑥} ∪ {𝑦}) ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
32	30, 31	sylan 281	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ ({𝑥} ∪ {𝑦}) ≈ 2o) → 1o ≈ 2o)
33	22, 32	mtand 654	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ¬ ({𝑥} ∪ {𝑦}) ≈ 2o)
34	33	necon2ai 2360	. . . . . . . . . 10 ⊢ (({𝑥} ∪ {𝑦}) ≈ 2o → 𝑥 ≠ 𝑦)
35		disjsn2 3581	. . . . . . . . . 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 3220	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∪ 𝐵) = ({𝑥} ∪ {𝑦}))
39	38	breq1d 3934	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o ↔ ({𝑥} ∪ {𝑦}) ≈ 2o))
40		ineq12 3267	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → (𝐴 ∩ 𝐵) = ({𝑥} ∩ {𝑦}))
41	40	eqeq1d 2146	. . . . . . . 8 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∩ 𝐵) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
42	37, 39, 41	3imtr4d 202	. . . . . . 7 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
43	42	ex 114	. . . . . 6 ⊢ (𝐴 = {𝑥} → (𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
44	43	exlimdv 1791	. . . . 5 ⊢ (𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
45	44	exlimiv 1577	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → (∃𝑦 𝐵 = {𝑦} → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅)))
46	45	imp 123	. . 3 ⊢ ((∃𝑥 𝐴 = {𝑥} ∧ ∃𝑦 𝐵 = {𝑦}) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
47	19, 20, 46	syl2anb 289	. 2 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∪ 𝐵) ≈ 2o → (𝐴 ∩ 𝐵) = ∅))
48	18, 47	impbid 128	1 ⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2306   ∪ cun 3064   ∩ cin 3065  ∅c0 3358  {csn 3522   class class class wbr 3924  Oncon0 4280  suc csuc 4282  1_oc1o 6299  2_oc2o 6300   ≈ cen 6625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1o 6306  df-2o 6307  df-er 6422  df-en 6628

This theorem is referenced by:  pr2nelem  7040  dju1p1e2  7046