Theorem ifr0 41154

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 2019	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5687	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 234	. . . 4 ⊢ 𝑥 I 𝑥
5		frirr 5496	. . . . 5 ⊢ (( I Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 416	. . . 4 ⊢ ( I Fr 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 139	. . 3 ⊢ ( I Fr 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 4312	. 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅)
9		fr0 5498	. . 3 ⊢ I Fr ∅
10		freq2 5490	. . 3 ⊢ (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
11	9, 10	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → I Fr 𝐴)
12	8, 11	impbii 212	1 ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∅c0 4243   class class class wbr 5030   I cid 5424   Fr wfr 5475

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-fr 5478  df-xp 5525  df-rel 5526

This theorem is referenced by: (None)