Theorem ifr0 43194

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 2015	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3478	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5850	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 230	. . . 4 ⊢ 𝑥 I 𝑥
5		frirr 5652	. . . . 5 ⊢ (( I Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 413	. . . 4 ⊢ ( I Fr 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 137	. . 3 ⊢ ( I Fr 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 4403	. 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅)
9		fr0 5654	. . 3 ⊢ I Fr ∅
10		freq2 5646	. . 3 ⊢ (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
11	9, 10	mpbiri 257	. 2 ⊢ (𝐴 = ∅ → I Fr 𝐴)
12	8, 11	impbii 208	1 ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∅c0 4321   class class class wbr 5147   I cid 5572   Fr wfr 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-fr 5630  df-xp 5681  df-rel 5682

This theorem is referenced by: (None)