Theorem uniintsn 4514

Description: Two ways to express "𝐴 is a singleton." See also en1 8023, en1b 8024, card1 8794, and eusn 4265. (Contributed by NM, 2-Aug-2010.)

Assertion

Ref	Expression
uniintsn	⊢ (∪ 𝐴 = ∩ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem uniintsn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vn0 3924	. . . . . 6 ⊢ V ≠ ∅
2		inteq 4478	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
3		int0 4490	. . . . . . . . . . 11 ⊢ ∩ ∅ = V
4	2, 3	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
5	4	adantl 482	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → ∩ 𝐴 = V)
6		unieq 4444	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
7		uni0 4465	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
8	6, 7	syl6eq 2672	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
9		eqeq1 2626	. . . . . . . . . . 11 ⊢ (∪ 𝐴 = ∩ 𝐴 → (∪ 𝐴 = ∅ ↔ ∩ 𝐴 = ∅))
10	8, 9	syl5ib 234	. . . . . . . . . 10 ⊢ (∪ 𝐴 = ∩ 𝐴 → (𝐴 = ∅ → ∩ 𝐴 = ∅))
11	10	imp 445	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → ∩ 𝐴 = ∅)
12	5, 11	eqtr3d 2658	. . . . . . . 8 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → V = ∅)
13	12	ex 450	. . . . . . 7 ⊢ (∪ 𝐴 = ∩ 𝐴 → (𝐴 = ∅ → V = ∅))
14	13	necon3d 2815	. . . . . 6 ⊢ (∪ 𝐴 = ∩ 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
15	1, 14	mpi 20	. . . . 5 ⊢ (∪ 𝐴 = ∩ 𝐴 → 𝐴 ≠ ∅)
16		n0 3931	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
17	15, 16	sylib 208	. . . 4 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
18		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
19		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
20	18, 19	prss 4351	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21		uniss 4458	. . . . . . . . . . . . 13 ⊢ ({𝑥, 𝑦} ⊆ 𝐴 → ∪ {𝑥, 𝑦} ⊆ ∪ 𝐴)
22	21	adantl 482	. . . . . . . . . . . 12 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∪ 𝐴)
23		simpl 473	. . . . . . . . . . . 12 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ 𝐴 = ∩ 𝐴)
24	22, 23	sseqtrd 3641	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∩ 𝐴)
25		intss 4498	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ⊆ 𝐴 → ∩ 𝐴 ⊆ ∩ {𝑥, 𝑦})
26	25	adantl 482	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∩ 𝐴 ⊆ ∩ {𝑥, 𝑦})
27	24, 26	sstrd 3613	. . . . . . . . . 10 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∩ {𝑥, 𝑦})
28	18, 19	unipr 4449	. . . . . . . . . 10 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
29	18, 19	intpr 4510	. . . . . . . . . 10 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦)
30	27, 28, 29	3sstr3g 3645	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦))
31		inss1 3833	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
32		ssun1 3776	. . . . . . . . . 10 ⊢ 𝑥 ⊆ (𝑥 ∪ 𝑦)
33	31, 32	sstri 3612	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)
34	30, 33	jctir 561	. . . . . . . 8 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ((𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)))
35		eqss 3618	. . . . . . . . 9 ⊢ ((𝑥 ∪ 𝑦) = (𝑥 ∩ 𝑦) ↔ ((𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)))
36		uneqin 3878	. . . . . . . . 9 ⊢ ((𝑥 ∪ 𝑦) = (𝑥 ∩ 𝑦) ↔ 𝑥 = 𝑦)
37	35, 36	bitr3i 266	. . . . . . . 8 ⊢ (((𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)) ↔ 𝑥 = 𝑦)
38	34, 37	sylib 208	. . . . . . 7 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
39	38	ex 450	. . . . . 6 ⊢ (∪ 𝐴 = ∩ 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴 → 𝑥 = 𝑦))
40	20, 39	syl5bi 232	. . . . 5 ⊢ (∪ 𝐴 = ∩ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
41	40	alrimivv 1856	. . . 4 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
42	17, 41	jca 554	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
43		euabsn 4261	. . . 4 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥})
44		eleq1 2689	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
45	44	eu4 2518	. . . 4 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
46		abid2 2745	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
47	46	eqeq1i 2627	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
48	47	exbii 1774	. . . 4 ⊢ (∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
49	43, 45, 48	3bitr3i 290	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
50	42, 49	sylib 208	. 2 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∃𝑥 𝐴 = {𝑥})
51	18	unisn 4451	. . . 4 ⊢ ∪ {𝑥} = 𝑥
52		unieq 4444	. . . 4 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
53		inteq 4478	. . . . 5 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
54	18	intsn 4513	. . . . 5 ⊢ ∩ {𝑥} = 𝑥
55	53, 54	syl6eq 2672	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
56	51, 52, 55	3eqtr4a 2682	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
57	56	exlimiv 1858	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
58	50, 57	impbii 199	1 ⊢ (∪ 𝐴 = ∩ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  {cab 2608   ≠ wne 2794  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ∪ cuni 4436  ∩ cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-int 4476

This theorem is referenced by:  uniintab  4515