Theorem ifr0 44446

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 2012	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5819	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 231	. . . 4 ⊢ 𝑥 I 𝑥
5		frirr 5617	. . . . 5 ⊢ (( I Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 412	. . . 4 ⊢ ( I Fr 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 137	. . 3 ⊢ ( I Fr 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 4373	. 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅)
9		fr0 5619	. . 3 ⊢ I Fr ∅
10		freq2 5609	. . 3 ⊢ (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅))
11	9, 10	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → I Fr 𝐴)
12	8, 11	impbii 209	1 ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∅c0 4299   class class class wbr 5110   I cid 5535   Fr wfr 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-fr 5594  df-xp 5647  df-rel 5648

This theorem is referenced by: (None)