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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 2009 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5866 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 231 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | frirr 5665 | . . . . 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 4413 | . 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅) |
9 | fr0 5667 | . . 3 ⊢ I Fr ∅ | |
10 | freq2 5657 | . . 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 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 I cid 5582 Fr wfr 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-fr 5641 df-xp 5695 df-rel 5696 |
This theorem is referenced by: (None) |
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