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Mirrors > Home > MPE Home > Th. List > ispod | Structured version Visualization version GIF version |
Description: Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.) |
Ref | Expression |
---|---|
ispod.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) |
ispod.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Ref | Expression |
---|---|
ispod | ⊢ (𝜑 → 𝑅 Po 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ispod.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) | |
2 | 1 | 3ad2antr1 1187 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ 𝑥𝑅𝑥) |
3 | ispod.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
4 | 2, 3 | jca 512 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
5 | 4 | ralrimivvva 3127 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
6 | df-po 5503 | . 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | |
7 | 5, 6 | sylibr 233 | 1 ⊢ (𝜑 → 𝑅 Po 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3064 class class class wbr 5074 Po wpo 5501 |
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 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1088 df-ral 3069 df-po 5503 |
This theorem is referenced by: swopo 5514 pofun 5521 issoi 5537 wemappo 9308 pospo 18063 legso 26960 pocnv 33730 poxp2 33790 poxp3 33796 lrrecpo 34098 |
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