![]() |
Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
|
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 2016 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3451 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5812 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 230 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | poirr 5561 | . . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥) | |
6 | 5 | ex 414 | . . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥)) |
7 | 4, 6 | mt2i 137 | . . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴) |
8 | 7 | eq0rdv 4368 | . 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅) |
9 | po0 5566 | . . 3 ⊢ I Po ∅ | |
10 | poeq2 5553 | . . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅)) | |
11 | 9, 10 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → I Po 𝐴) |
12 | 8, 11 | impbii 208 | 1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∅c0 4286 class class class wbr 5109 I cid 5534 Po wpo 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-po 5549 df-xp 5643 df-rel 5644 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |