Theorem frsn 5361

Description: Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
frsn	⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem frsn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snprc 4410	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		fr0 5258	. . . . . . 7 ⊢ 𝑅 Fr ∅
3		freq2 5250	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 Fr {𝐴} ↔ 𝑅 Fr ∅))
4	2, 3	mpbiri 249	. . . . . 6 ⊢ ({𝐴} = ∅ → 𝑅 Fr {𝐴})
5	1, 4	sylbi 208	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝑅 Fr {𝐴})
6	5	adantl 473	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Fr {𝐴})
7		brrelex1 5327	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐴) → 𝐴 ∈ V)
8	7	stoic1a 1867	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
9	6, 8	2thd 256	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
10	9	ex 401	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11		df-fr 5238	. . . 4 ⊢ (𝑅 Fr {𝐴} ↔ ∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
12		sssn 4513	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
13		neor 3028	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ↔ (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
14	12, 13	sylbb 210	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
15	14	imp 395	. . . . . . . . 9 ⊢ ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
16	15	adantl 473	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
17		eqimss 3819	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
18	17	adantl 473	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
19		snnzg 4464	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → {𝐴} ≠ ∅)
20		neeq1 2999	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴} → (𝑥 ≠ ∅ ↔ {𝐴} ≠ ∅))
21	19, 20	syl5ibrcom 238	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 = {𝐴} → 𝑥 ≠ ∅))
22	21	imp 395	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ≠ ∅)
23	18, 22	jca 507	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅))
24	16, 23	impbida 835	. . . . . . 7 ⊢ (𝐴 ∈ V → ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) ↔ 𝑥 = {𝐴}))
25	24	imbi1d 332	. . . . . 6 ⊢ (𝐴 ∈ V → (((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)))
26	25	albidv 2015	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)))
27		snex 5066	. . . . . 6 ⊢ {𝐴} ∈ V
28		raleq 3286	. . . . . . 7 ⊢ (𝑥 = {𝐴} → (∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
29	28	rexeqbi1dv 3295	. . . . . 6 ⊢ (𝑥 = {𝐴} → (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
30	27, 29	ceqsalv 3386	. . . . 5 ⊢ (∀𝑥(𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦)
31	26, 30	syl6bb 278	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
32	11, 31	syl5bb 274	. . 3 ⊢ (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
33		breq2 4815	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝐴))
34	33	notbid 309	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐴))
35	34	ralbidv 3133	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
36	35	rexsng 4378	. . 3 ⊢ (𝐴 ∈ V → (∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
37		breq1 4814	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧𝑅𝐴 ↔ 𝐴𝑅𝐴))
38	37	notbid 309	. . . 4 ⊢ (𝑧 = 𝐴 → (¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
39	38	ralsng 4377	. . 3 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
40	32, 36, 39	3bitrd 296	. 2 ⊢ (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
41	10, 40	pm2.61d2 173	1 ⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 873  ∀wal 1650   = wceq 1652   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055  ∃wrex 3056  Vcvv 3350   ⊆ wss 3734  ∅c0 4081  {csn 4336   class class class wbr 4811   Fr wfr 5235  Rel wrel 5284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-fr 5238  df-xp 5285  df-rel 5286

This theorem is referenced by:  wesn  5362