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Mirrors > Home > ILE Home > Th. List > po0 | GIF version |
Description: Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
po0 | ⊢ 𝑅 Po ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 3525 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
2 | df-po 4297 | . 2 ⊢ (𝑅 Po ∅ ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | |
3 | 1, 2 | mpbir 146 | 1 ⊢ 𝑅 Po ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∀wral 2455 ∅c0 3423 class class class wbr 4004 Po wpo 4295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2740 df-dif 3132 df-nul 3424 df-po 4297 |
This theorem is referenced by: so0 4327 |
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