Theorem frsn 5603

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 4613	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		fr0 5498	. . . . . . 7 ⊢ 𝑅 Fr ∅
3		freq2 5490	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 Fr {𝐴} ↔ 𝑅 Fr ∅))
4	2, 3	mpbiri 261	. . . . . 6 ⊢ ({𝐴} = ∅ → 𝑅 Fr {𝐴})
5	1, 4	sylbi 220	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝑅 Fr {𝐴})
6	5	adantl 485	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Fr {𝐴})
7		brrelex1 5569	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐴) → 𝐴 ∈ V)
8	7	stoic1a 1774	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
9	6, 8	2thd 268	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
10	9	ex 416	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11		df-fr 5478	. . . 4 ⊢ (𝑅 Fr {𝐴} ↔ ∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
12		sssn 4719	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
13		neor 3078	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ↔ (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
14	12, 13	sylbb 222	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
15	14	imp 410	. . . . . . . . 9 ⊢ ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
16	15	adantl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
17		eqimss 3971	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
18	17	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
19		snnzg 4670	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → {𝐴} ≠ ∅)
20		neeq1 3049	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴} → (𝑥 ≠ ∅ ↔ {𝐴} ≠ ∅))
21	19, 20	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 = {𝐴} → 𝑥 ≠ ∅))
22	21	imp 410	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ≠ ∅)
23	18, 22	jca 515	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅))
24	16, 23	impbida 800	. . . . . . 7 ⊢ (𝐴 ∈ V → ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) ↔ 𝑥 = {𝐴}))
25	24	imbi1d 345	. . . . . 6 ⊢ (𝐴 ∈ V → (((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)))
26	25	albidv 1921	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)))
27		snex 5297	. . . . . 6 ⊢ {𝐴} ∈ V
28		raleq 3358	. . . . . . 7 ⊢ (𝑥 = {𝐴} → (∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
29	28	rexeqbi1dv 3357	. . . . . 6 ⊢ (𝑥 = {𝐴} → (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
30	27, 29	ceqsalv 3479	. . . . 5 ⊢ (∀𝑥(𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦)
31	26, 30	syl6bb 290	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
32	11, 31	syl5bb 286	. . 3 ⊢ (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
33		breq2 5034	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝐴))
34	33	notbid 321	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐴))
35	34	ralbidv 3162	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
36	35	rexsng 4574	. . 3 ⊢ (𝐴 ∈ V → (∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
37		breq1 5033	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧𝑅𝐴 ↔ 𝐴𝑅𝐴))
38	37	notbid 321	. . . 4 ⊢ (𝑧 = 𝐴 → (¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
39	38	ralsng 4573	. . 3 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
40	32, 36, 39	3bitrd 308	. 2 ⊢ (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
41	10, 40	pm2.61d2 184	1 ⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5030   Fr wfr 5475  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-fr 5478  df-xp 5525  df-rel 5526

This theorem is referenced by:  wesn  5604