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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 4075 | . 2 ⊢ Tr ∅ | |
2 | ral0 3496 | . 2 ⊢ ∀𝑥 ∈ ∅ Tr 𝑥 | |
3 | dford3 4329 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ ∀𝑥 ∈ ∅ Tr 𝑥)) | |
4 | 1, 2, 3 | mpbir2an 927 | 1 ⊢ Ord ∅ |
Colors of variables: wff set class |
Syntax hints: ∀wral 2435 ∅c0 3395 Tr wtr 4064 Ord word 4324 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-uni 3775 df-tr 4065 df-iord 4328 |
This theorem is referenced by: 0elon 4354 ordtriexmidlem 4480 2ordpr 4485 smo0 6247 |
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