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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 2010 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3495 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5716 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 232 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | poirr 5478 | . . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥) | |
6 | 5 | ex 413 | . . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥)) |
7 | 4, 6 | mt2i 139 | . . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴) |
8 | 7 | eq0rdv 4354 | . 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅) |
9 | po0 5483 | . . 3 ⊢ I Po ∅ | |
10 | poeq2 5471 | . . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅)) | |
11 | 9, 10 | mpbiri 259 | . 2 ⊢ (𝐴 = ∅ → I Po 𝐴) |
12 | 8, 11 | impbii 210 | 1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∅c0 4288 class class class wbr 5057 I cid 5452 Po wpo 5465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-po 5467 df-xp 5554 df-rel 5555 |
This theorem is referenced by: (None) |
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