Theorem frsn 5726

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 4681	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		fr0 5616	. . . . . . 7 ⊢ 𝑅 Fr ∅
3		freq2 5606	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 Fr {𝐴} ↔ 𝑅 Fr ∅))
4	2, 3	mpbiri 258	. . . . . 6 ⊢ ({𝐴} = ∅ → 𝑅 Fr {𝐴})
5	1, 4	sylbi 217	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝑅 Fr {𝐴})
6	5	adantl 481	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Fr {𝐴})
7		brrelex1 5691	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐴) → 𝐴 ∈ V)
8	7	stoic1a 1772	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
9	6, 8	2thd 265	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
10	9	ex 412	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11		df-fr 5591	. . . 4 ⊢ (𝑅 Fr {𝐴} ↔ ∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
12		sssn 4790	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
13		neor 3017	. . . . . . . . . . 11 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ↔ (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
14	12, 13	sylbb 219	. . . . . . . . . 10 ⊢ (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
15	14	imp 406	. . . . . . . . 9 ⊢ ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
17		eqimss 4005	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
18	17	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
19		snnzg 4738	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → {𝐴} ≠ ∅)
20		neeq1 2987	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴} → (𝑥 ≠ ∅ ↔ {𝐴} ≠ ∅))
21	19, 20	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 = {𝐴} → 𝑥 ≠ ∅))
22	21	imp 406	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ≠ ∅)
23	18, 22	jca 511	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅))
24	16, 23	impbida 800	. . . . . . 7 ⊢ (𝐴 ∈ V → ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) ↔ 𝑥 = {𝐴}))
25	24	imbi1d 341	. . . . . 6 ⊢ (𝐴 ∈ V → (((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)))
26	25	albidv 1920	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)))
27		snex 5391	. . . . . 6 ⊢ {𝐴} ∈ V
28		raleq 3296	. . . . . . 7 ⊢ (𝑥 = {𝐴} → (∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
29	28	rexeqbi1dv 3312	. . . . . 6 ⊢ (𝑥 = {𝐴} → (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
30	27, 29	ceqsalv 3487	. . . . 5 ⊢ (∀𝑥(𝑥 = {𝐴} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦)
31	26, 30	bitrdi 287	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
32	11, 31	bitrid 283	. . 3 ⊢ (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
33		breq2 5111	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝐴))
34	33	notbid 318	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐴))
35	34	ralbidv 3156	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
36	35	rexsng 4640	. . 3 ⊢ (𝐴 ∈ V → (∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
37		breq1 5110	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧𝑅𝐴 ↔ 𝐴𝑅𝐴))
38	37	notbid 318	. . . 4 ⊢ (𝑧 = 𝐴 → (¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
39	38	ralsng 4639	. . 3 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
40	32, 36, 39	3bitrd 305	. 2 ⊢ (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
41	10, 40	pm2.61d2 181	1 ⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  {csn 4589   class class class wbr 5107   Fr wfr 5588  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-fr 5591  df-xp 5644  df-rel 5645

This theorem is referenced by:  wesn  5727