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Theorem frsn 5724
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 4683 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
2 fr0 5617 . . . . . . 7 𝑅 Fr ∅
3 freq2 5609 . . . . . . 7 ({𝐴} = ∅ → (𝑅 Fr {𝐴} ↔ 𝑅 Fr ∅))
42, 3mpbiri 258 . . . . . 6 ({𝐴} = ∅ → 𝑅 Fr {𝐴})
51, 4sylbi 216 . . . . 5 𝐴 ∈ V → 𝑅 Fr {𝐴})
65adantl 483 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Fr {𝐴})
7 brrelex1 5690 . . . . 5 ((Rel 𝑅𝐴𝑅𝐴) → 𝐴 ∈ V)
87stoic1a 1775 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
96, 82thd 265 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
109ex 414 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11 df-fr 5593 . . . 4 (𝑅 Fr {𝐴} ↔ ∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
12 sssn 4791 . . . . . . . . . . 11 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
13 neor 3037 . . . . . . . . . . 11 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ↔ (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1412, 13sylbb 218 . . . . . . . . . 10 (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → 𝑥 = {𝐴}))
1514imp 408 . . . . . . . . 9 ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → 𝑥 = {𝐴})
1615adantl 483 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → 𝑥 = {𝐴})
17 eqimss 4005 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
1817adantl 483 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
19 snnzg 4740 . . . . . . . . . . 11 (𝐴 ∈ V → {𝐴} ≠ ∅)
20 neeq1 3007 . . . . . . . . . . 11 (𝑥 = {𝐴} → (𝑥 ≠ ∅ ↔ {𝐴} ≠ ∅))
2119, 20syl5ibrcom 247 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 = {𝐴} → 𝑥 ≠ ∅))
2221imp 408 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → 𝑥 ≠ ∅)
2318, 22jca 513 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥 = {𝐴}) → (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅))
2416, 23impbida 800 . . . . . . 7 (𝐴 ∈ V → ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) ↔ 𝑥 = {𝐴}))
2524imbi1d 342 . . . . . 6 (𝐴 ∈ V → (((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
2625albidv 1924 . . . . 5 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
27 snex 5393 . . . . . 6 {𝐴} ∈ V
28 raleq 3312 . . . . . . 7 (𝑥 = {𝐴} → (∀𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
2928rexeqbi1dv 3311 . . . . . 6 (𝑥 = {𝐴} → (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3027, 29ceqsalv 3484 . . . . 5 (∀𝑥(𝑥 = {𝐴} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦)
3126, 30bitrdi 287 . . . 4 (𝐴 ∈ V → (∀𝑥((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
3211, 31bitrid 283 . . 3 (𝐴 ∈ V → (𝑅 Fr {𝐴} ↔ ∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦))
33 breq2 5114 . . . . . 6 (𝑦 = 𝐴 → (𝑧𝑅𝑦𝑧𝑅𝐴))
3433notbid 318 . . . . 5 (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐴))
3534ralbidv 3175 . . . 4 (𝑦 = 𝐴 → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
3635rexsng 4640 . . 3 (𝐴 ∈ V → (∃𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴))
37 breq1 5113 . . . . 5 (𝑧 = 𝐴 → (𝑧𝑅𝐴𝐴𝑅𝐴))
3837notbid 318 . . . 4 (𝑧 = 𝐴 → (¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
3938ralsng 4639 . . 3 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝐴 ↔ ¬ 𝐴𝑅𝐴))
4032, 36, 393bitrd 305 . 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 397  wo 846  wal 1540   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  wss 3915  c0 4287  {csn 4591   class class class wbr 5110   Fr wfr 5590  Rel wrel 5643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-fr 5593  df-xp 5644  df-rel 5645
This theorem is referenced by:  wesn  5725
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