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Theorem frsn 5344
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 4395 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
2 fr0 5243 . . . . . . 7 𝑅 Fr ∅
3 freq2 5235 . . . . . . 7 ({𝐴} = ∅ → (𝑅 Fr {𝐴} ↔ 𝑅 Fr ∅))
42, 3mpbiri 248 . . . . . 6 ({𝐴} = ∅ → 𝑅 Fr {𝐴})
51, 4sylbi 207 . . . . 5 𝐴 ∈ V → 𝑅 Fr {𝐴})
65adantl 473 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Fr {𝐴})
7 brrelex 5311 . . . . 5 ((Rel 𝑅𝐴𝑅𝐴) → 𝐴 ∈ V)
87stoic1a 1844 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
96, 82thd 255 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
109ex 449 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11 df-fr 5223 . . . 4 (𝑅 Fr {𝐴} ↔ ∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
12 sssn 4501 . . . . . . . . . . 11 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
13 neor 3021 . . . . . . . . . . 11 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ↔ (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1412, 13sylbb 209 . . . . . . . . . 10 (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1514imp 444 . . . . . . . . 9 ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
1615adantl 473 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
17 eqimss 3796 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
1817adantl 473 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
19 snnzg 4449 . . . . . . . . . . 11 (𝐴 ∈ V → {𝐴} ≠ ∅)
20 neeq1 2992 . . . . . . . . . . 11 (𝑥 = {𝐴} → (𝑥 ≠ ∅ ↔ {𝐴} ≠ ∅))
2119, 20syl5ibrcom 237 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 = {𝐴} → 𝑥 ≠ ∅))
2221imp 444 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ≠ ∅)
2318, 22jca 555 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅))
2416, 23impbida 913 . . . . . . 7 (𝐴 ∈ V → ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) ↔ 𝑥 = {𝐴}))
2524imbi1d 330 . . . . . 6 (𝐴 ∈ V → (((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
2625albidv 1996 . . . . 5 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
27 snex 5055 . . . . . 6 {𝐴} ∈ V
28 raleq 3275 . . . . . . 7 (𝑥 = {𝐴} → (∀𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
2928rexeqbi1dv 3284 . . . . . 6 (𝑥 = {𝐴} → (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3027, 29ceqsalv 3371 . . . . 5 (∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦)
3126, 30syl6bb 276 . . . 4 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3211, 31syl5bb 272 . . 3 (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
33 breq2 4806 . . . . . 6 (𝑦 = 𝐴 → (𝑧𝑅𝑦𝑧𝑅𝐴))
3433notbid 307 . . . . 5 (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐴))
3534ralbidv 3122 . . . 4 (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
3635rexsng 4361 . . 3 (𝐴 ∈ V → (∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
37 breq1 4805 . . . . 5 (𝑧 = 𝐴 → (𝑧𝑅𝐴𝐴𝑅𝐴))
3837notbid 307 . . . 4 (𝑧 = 𝐴 → (¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
3938ralsng 4360 . . 3 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
4032, 36, 393bitrd 294 . 2 (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
4110, 40pm2.61d2 172 1 (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wal 1628   = wceq 1630  wcel 2137  wne 2930  wral 3048  wrex 3049  Vcvv 3338  wss 3713  c0 4056  {csn 4319   class class class wbr 4802   Fr wfr 5220  Rel wrel 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-fr 5223  df-xp 5270  df-rel 5271
This theorem is referenced by:  wesn  5345
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