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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 2011 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | 2 | ideq 5863 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
| 4 | 1, 3 | mpbir 231 | . . . 4 ⊢ 𝑥 I 𝑥 |
| 5 | frirr 5661 | . . . . 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 4407 | . 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅) |
| 9 | fr0 5663 | . . 3 ⊢ I Fr ∅ | |
| 10 | freq2 5653 | . . 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 1540 ∈ wcel 2108 ∅c0 4333 class class class wbr 5143 I cid 5577 Fr wfr 5634 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-fr 5637 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: (None) |
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