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Theorem ifr0 44446
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 2009 . . . . 5 𝑥 = 𝑥
2 vex 3482 . . . . . 6 𝑥 ∈ V
32ideq 5866 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 231 . . . 4 𝑥 I 𝑥
5 frirr 5665 . . . . 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 4413 . 2 ( I Fr 𝐴𝐴 = ∅)
9 fr0 5667 . . 3 I Fr ∅
10 freq2 5657 . . 3 (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
119, 10mpbiri 258 . 2 (𝐴 = ∅ → I Fr 𝐴)
128, 11impbii 209 1 ( I Fr 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wcel 2106  c0 4339   class class class wbr 5148   I cid 5582   Fr wfr 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-fr 5641  df-xp 5695  df-rel 5696
This theorem is referenced by: (None)
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