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| Description: The empty set is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 11-May-1994.) | 
| Ref | Expression | 
|---|---|
| ord0 | ⊢ Ord ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tr0 5272 | . 2 ⊢ Tr ∅ | |
| 2 | we0 5680 | . 2 ⊢ E We ∅ | |
| 3 | df-ord 6387 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ E We ∅)) | |
| 4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ Ord ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∅c0 4333 Tr wtr 5259 E cep 5583 We wwe 5636 Ord word 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 | 
| This theorem is referenced by: 0elon 6438 ord0eln0 6439 ordzsl 7866 smo0 8398 oicl 9569 alephgeom 10122 pw2bday 28418 | 
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