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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 2094 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3343 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5430 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 221 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | poirr 5198 | . . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥) | |
6 | 5 | ex 449 | . . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥)) |
7 | 4, 6 | mt2i 132 | . . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴) |
8 | 7 | eq0rdv 4122 | . 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅) |
9 | po0 5202 | . . 3 ⊢ I Po ∅ | |
10 | poeq2 5191 | . . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅)) | |
11 | 9, 10 | mpbiri 248 | . 2 ⊢ (𝐴 = ∅ → I Po 𝐴) |
12 | 8, 11 | impbii 199 | 1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1632 ∈ wcel 2139 ∅c0 4058 class class class wbr 4804 I cid 5173 Po wpo 5185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-id 5174 df-po 5187 df-xp 5272 df-rel 5273 |
This theorem is referenced by: (None) |
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