![]() |
Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ifr0 | Structured version Visualization version GIF version |
Description: A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ifr0 | ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2008 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3475 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5855 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 230 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | frirr 5655 | . . . . 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 4405 | . 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅) |
9 | fr0 5657 | . . 3 ⊢ I Fr ∅ | |
10 | freq2 5649 | . . 3 ⊢ (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅)) | |
11 | 9, 10 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → I Fr 𝐴) |
12 | 8, 11 | impbii 208 | 1 ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∅c0 4323 class class class wbr 5148 I cid 5575 Fr wfr 5630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-fr 5633 df-xp 5684 df-rel 5685 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |