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Theorem ifr0 43723
Description: A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ifr0 ( I Fr 𝐴𝐴 = ∅)

Proof of Theorem ifr0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 equid 2007 . . . . 5 𝑥 = 𝑥
2 vex 3470 . . . . . 6 𝑥 ∈ V
32ideq 5843 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 230 . . . 4 𝑥 I 𝑥
5 frirr 5644 . . . . 5 (( I Fr 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 412 . . . 4 ( I Fr 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 137 . . 3 ( I Fr 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 4397 . 2 ( I Fr 𝐴𝐴 = ∅)
9 fr0 5646 . . 3 I Fr ∅
10 freq2 5638 . . 3 (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
119, 10mpbiri 258 . 2 (𝐴 = ∅ → I Fr 𝐴)
128, 11impbii 208 1 ( I Fr 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  c0 4315   class class class wbr 5139   I cid 5564   Fr wfr 5619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-fr 5622  df-xp 5673  df-rel 5674
This theorem is referenced by: (None)
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