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Theorem frsn 5760
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 4717 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
2 fr0 5652 . . . . . . 7 𝑅 Fr ∅
3 freq2 5644 . . . . . . 7 ({𝐴} = ∅ → (𝑅 Fr {𝐴} ↔ 𝑅 Fr ∅))
42, 3mpbiri 257 . . . . . 6 ({𝐴} = ∅ → 𝑅 Fr {𝐴})
51, 4sylbi 216 . . . . 5 𝐴 ∈ V → 𝑅 Fr {𝐴})
65adantl 480 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Fr {𝐴})
7 brrelex1 5726 . . . . 5 ((Rel 𝑅𝐴𝑅𝐴) → 𝐴 ∈ V)
87stoic1a 1767 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
96, 82thd 264 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
109ex 411 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11 df-fr 5628 . . . 4 (𝑅 Fr {𝐴} ↔ ∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
12 sssn 4826 . . . . . . . . . . 11 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
13 neor 3024 . . . . . . . . . . 11 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ↔ (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1412, 13sylbb 218 . . . . . . . . . 10 (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1514imp 405 . . . . . . . . 9 ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
1615adantl 480 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
17 eqimss 4038 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
1817adantl 480 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
19 snnzg 4774 . . . . . . . . . . 11 (𝐴 ∈ V → {𝐴} ≠ ∅)
20 neeq1 2993 . . . . . . . . . . 11 (𝑥 = {𝐴} → (𝑥 ≠ ∅ ↔ {𝐴} ≠ ∅))
2119, 20syl5ibrcom 246 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 = {𝐴} → 𝑥 ≠ ∅))
2221imp 405 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ≠ ∅)
2318, 22jca 510 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅))
2416, 23impbida 799 . . . . . . 7 (𝐴 ∈ V → ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) ↔ 𝑥 = {𝐴}))
2524imbi1d 340 . . . . . 6 (𝐴 ∈ V → (((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
2625albidv 1916 . . . . 5 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
27 snex 5428 . . . . . 6 {𝐴} ∈ V
28 raleq 3312 . . . . . . 7 (𝑥 = {𝐴} → (∀𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
2928rexeqbi1dv 3324 . . . . . 6 (𝑥 = {𝐴} → (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3027, 29ceqsalv 3503 . . . . 5 (∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦)
3126, 30bitrdi 286 . . . 4 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3211, 31bitrid 282 . . 3 (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
33 breq2 5148 . . . . . 6 (𝑦 = 𝐴 → (𝑧𝑅𝑦𝑧𝑅𝐴))
3433notbid 317 . . . . 5 (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐴))
3534ralbidv 3168 . . . 4 (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
3635rexsng 4674 . . 3 (𝐴 ∈ V → (∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
37 breq1 5147 . . . . 5 (𝑧 = 𝐴 → (𝑧𝑅𝐴𝐴𝑅𝐴))
3837notbid 317 . . . 4 (𝑧 = 𝐴 → (¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
3938ralsng 4673 . . 3 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
4032, 36, 393bitrd 304 . 2 (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
4110, 40pm2.61d2 181 1 (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  wal 1532   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3463  wss 3947  c0 4323  {csn 4624   class class class wbr 5144   Fr wfr 5625  Rel wrel 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-fr 5628  df-xp 5679  df-rel 5680
This theorem is referenced by:  wesn  5761
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