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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 4098 | . 2 ⊢ Tr ∅ | |
2 | ral0 3516 | . 2 ⊢ ∀𝑥 ∈ ∅ Tr 𝑥 | |
3 | dford3 4352 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ ∀𝑥 ∈ ∅ Tr 𝑥)) | |
4 | 1, 2, 3 | mpbir2an 937 | 1 ⊢ Ord ∅ |
Colors of variables: wff set class |
Syntax hints: ∀wral 2448 ∅c0 3414 Tr wtr 4087 Ord word 4347 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-uni 3797 df-tr 4088 df-iord 4351 |
This theorem is referenced by: 0elon 4377 ordtriexmidlem 4503 2ordpr 4508 smo0 6277 |
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