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 𝐴 ↔ 𝐴 = ∅)

Proof of Theorem ifr0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 1938	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3201	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5272	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 221	. . . 4 ⊢ 𝑥 I 𝑥
5		frirr 5089	. . . . 5 ⊢ (( I Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 450	. . . 4 ⊢ ( I Fr 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 132	. . 3 ⊢ ( I Fr 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 3977	. 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅)
9		fr0 5091	. . 3 ⊢ I Fr ∅
10		freq2 5083	. . 3 ⊢ (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
11	9, 10	mpbiri 248	. 2 ⊢ (𝐴 = ∅ → I Fr 𝐴)
12	8, 11	impbii 199	1 ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅)

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)