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Mirrors > Home > ILE Home > Th. List > ord0 | GIF version |
Description: The empty set is an ordinal class. (Contributed by NM, 11-May-1994.) |
Ref | Expression |
---|---|
ord0 | ⊢ Ord ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tr0 4037 | . 2 ⊢ Tr ∅ | |
2 | ral0 3464 | . 2 ⊢ ∀𝑥 ∈ ∅ Tr 𝑥 | |
3 | dford3 4289 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ ∀𝑥 ∈ ∅ Tr 𝑥)) | |
4 | 1, 2, 3 | mpbir2an 926 | 1 ⊢ Ord ∅ |
Colors of variables: wff set class |
Syntax hints: ∀wral 2416 ∅c0 3363 Tr wtr 4026 Ord word 4284 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-uni 3737 df-tr 4027 df-iord 4288 |
This theorem is referenced by: 0elon 4314 ordtriexmidlem 4435 2ordpr 4439 smo0 6195 |
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