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Theorem ifr0 39434
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 2111 . . . . 5 𝑥 = 𝑥
2 vex 3388 . . . . . 6 𝑥 ∈ V
32ideq 5478 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 223 . . . 4 𝑥 I 𝑥
5 frirr 5289 . . . . 5 (( I Fr 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 402 . . . 4 ( I Fr 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 135 . . 3 ( I Fr 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 4175 . 2 ( I Fr 𝐴𝐴 = ∅)
9 fr0 5291 . . 3 I Fr ∅
10 freq2 5283 . . 3 (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
119, 10mpbiri 250 . 2 (𝐴 = ∅ → I Fr 𝐴)
128, 11impbii 201 1 ( I Fr 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198   = wceq 1653  wcel 2157  c0 4115   class class class wbr 4843   I cid 5219   Fr wfr 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-fr 5271  df-xp 5318  df-rel 5319
This theorem is referenced by: (None)
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