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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ipo0 | Structured version Visualization version GIF version | ||
| Description: If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.) |
| Ref | Expression |
|---|---|
| ipo0 | ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equid 2011 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
| 2 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | 2 | ideq 5861 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
| 4 | 1, 3 | mpbir 231 | . . . 4 ⊢ 𝑥 I 𝑥 |
| 5 | poirr 5602 | . . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥) | |
| 6 | 5 | ex 412 | . . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥)) |
| 7 | 4, 6 | mt2i 137 | . . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴) |
| 8 | 7 | eq0rdv 4406 | . 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅) |
| 9 | po0 5607 | . . 3 ⊢ I Po ∅ | |
| 10 | poeq2 5594 | . . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅)) | |
| 11 | 9, 10 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → I Po 𝐴) |
| 12 | 8, 11 | impbii 209 | 1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∅c0 4332 class class class wbr 5141 I cid 5575 Po wpo 5588 |
| 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 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| 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 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5142 df-opab 5204 df-id 5576 df-po 5590 df-xp 5689 df-rel 5690 |
| This theorem is referenced by: (None) |
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