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Theorem ifr0 44469
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 2011 . . . . 5 𝑥 = 𝑥
2 vex 3484 . . . . . 6 𝑥 ∈ V
32ideq 5863 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 231 . . . 4 𝑥 I 𝑥
5 frirr 5661 . . . . 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 4407 . 2 ( I Fr 𝐴𝐴 = ∅)
9 fr0 5663 . . 3 I Fr ∅
10 freq2 5653 . . 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 1540  wcel 2108  c0 4333   class class class wbr 5143   I cid 5577   Fr wfr 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-fr 5637  df-xp 5691  df-rel 5692
This theorem is referenced by: (None)
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