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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.
StepHypRef Expression
1 snprc 4410 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
2 fr0 5258 . . . . . . 7 𝑅 Fr ∅
3 freq2 5250 . . . . . . 7 ({𝐴} = ∅ → (𝑅 Fr {𝐴} ↔ 𝑅 Fr ∅))
42, 3mpbiri 249 . . . . . 6 ({𝐴} = ∅ → 𝑅 Fr {𝐴})
51, 4sylbi 208 . . . . 5 𝐴 ∈ V → 𝑅 Fr {𝐴})
65adantl 473 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Fr {𝐴})
7 brrelex1 5327 . . . . 5 ((Rel 𝑅𝐴𝑅𝐴) → 𝐴 ∈ V)
87stoic1a 1867 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
96, 82thd 256 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
109ex 401 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11 df-fr 5238 . . . 4 (𝑅 Fr {𝐴} ↔ ∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
12 sssn 4513 . . . . . . . . . . 11 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
13 neor 3028 . . . . . . . . . . 11 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ↔ (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1412, 13sylbb 210 . . . . . . . . . 10 (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1514imp 395 . . . . . . . . 9 ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
1615adantl 473 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
17 eqimss 3819 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
1817adantl 473 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
19 snnzg 4464 . . . . . . . . . . 11 (𝐴 ∈ V → {𝐴} ≠ ∅)
20 neeq1 2999 . . . . . . . . . . 11 (𝑥 = {𝐴} → (𝑥 ≠ ∅ ↔ {𝐴} ≠ ∅))
2119, 20syl5ibrcom 238 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 = {𝐴} → 𝑥 ≠ ∅))
2221imp 395 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ≠ ∅)
2318, 22jca 507 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅))
2416, 23impbida 835 . . . . . . 7 (𝐴 ∈ V → ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) ↔ 𝑥 = {𝐴}))
2524imbi1d 332 . . . . . 6 (𝐴 ∈ V → (((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
2625albidv 2015 . . . . 5 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
27 snex 5066 . . . . . 6 {𝐴} ∈ V
28 raleq 3286 . . . . . . 7 (𝑥 = {𝐴} → (∀𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
2928rexeqbi1dv 3295 . . . . . 6 (𝑥 = {𝐴} → (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3027, 29ceqsalv 3386 . . . . 5 (∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦)
3126, 30syl6bb 278 . . . 4 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3211, 31syl5bb 274 . . 3 (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
33 breq2 4815 . . . . . 6 (𝑦 = 𝐴 → (𝑧𝑅𝑦𝑧𝑅𝐴))
3433notbid 309 . . . . 5 (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐴))
3534ralbidv 3133 . . . 4 (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
3635rexsng 4378 . . 3 (𝐴 ∈ V → (∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
37 breq1 4814 . . . . 5 (𝑧 = 𝐴 → (𝑧𝑅𝐴𝐴𝑅𝐴))
3837notbid 309 . . . 4 (𝑧 = 𝐴 → (¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
3938ralsng 4377 . . 3 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
4032, 36, 393bitrd 296 . 2 (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
4110, 40pm2.61d2 173 1 (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
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
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