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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 2007 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3481 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5861 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 231 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | poirr 5604 | . . . . 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 4413 | . 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅) |
9 | po0 5609 | . . 3 ⊢ I Po ∅ | |
10 | poeq2 5595 | . . 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 1535 ∈ wcel 2104 ∅c0 4339 class class class wbr 5150 I cid 5576 Po wpo 5589 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5151 df-opab 5213 df-id 5577 df-po 5591 df-xp 5690 df-rel 5691 |
This theorem is referenced by: (None) |
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