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Theorem ifr0 38480
 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 1938 . . . . 5 𝑥 = 𝑥
2 vex 3201 . . . . . 6 𝑥 ∈ V
32ideq 5272 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 221 . . . 4 𝑥 I 𝑥
5 frirr 5089 . . . . 5 (( I Fr 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 450 . . . 4 ( I Fr 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 132 . . 3 ( I Fr 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 3977 . 2 ( I Fr 𝐴𝐴 = ∅)
9 fr0 5091 . . 3 I Fr ∅
10 freq2 5083 . . 3 (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
119, 10mpbiri 248 . 2 (𝐴 = ∅ → I Fr 𝐴)
128, 11impbii 199 1 ( I Fr 𝐴𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1482   ∈ wcel 1989  ∅c0 3913   class class class wbr 4651   I cid 5021   Fr wfr 5068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-id 5022  df-fr 5071  df-xp 5118  df-rel 5119 This theorem is referenced by: (None)
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