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Mirrors > Home > ILE Home > Th. List > we0 | GIF version |
Description: Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.) |
Ref | Expression |
---|---|
we0 | ⊢ 𝑅 We ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fr0 4281 | . 2 ⊢ 𝑅 Fr ∅ | |
2 | ral0 3469 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) | |
3 | df-wetr 4264 | . 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | |
4 | 1, 2, 3 | mpbir2an 927 | 1 ⊢ 𝑅 We ∅ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2417 ∅c0 3368 class class class wbr 3937 Fr wfr 4258 We wwe 4260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-frfor 4261 df-frind 4262 df-wetr 4264 |
This theorem is referenced by: (None) |
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