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Theorem ifr0 43881
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 2008 . . . . 5 𝑥 = 𝑥
2 vex 3474 . . . . . 6 𝑥 ∈ V
32ideq 5849 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 230 . . . 4 𝑥 I 𝑥
5 frirr 5649 . . . . 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 4400 . 2 ( I Fr 𝐴𝐴 = ∅)
9 fr0 5651 . . 3 I Fr ∅
10 freq2 5643 . . 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 1534  wcel 2099  c0 4318   class class class wbr 5142   I cid 5569   Fr wfr 5624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-fr 5627  df-xp 5678  df-rel 5679
This theorem is referenced by: (None)
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