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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 4007 | . 2 ⊢ Tr ∅ | |
2 | ral0 3434 | . 2 ⊢ ∀𝑥 ∈ ∅ Tr 𝑥 | |
3 | dford3 4259 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ ∀𝑥 ∈ ∅ Tr 𝑥)) | |
4 | 1, 2, 3 | mpbir2an 911 | 1 ⊢ Ord ∅ |
Colors of variables: wff set class |
Syntax hints: ∀wral 2393 ∅c0 3333 Tr wtr 3996 Ord word 4254 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-uni 3707 df-tr 3997 df-iord 4258 |
This theorem is referenced by: 0elon 4284 ordtriexmidlem 4405 2ordpr 4409 smo0 6163 |
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