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Mirrors > Home > MPE Home > Th. List > ord0 | Structured version Visualization version GIF version |
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 5279 | . 2 ⊢ Tr ∅ | |
2 | we0 5673 | . 2 ⊢ E We ∅ | |
3 | df-ord 6374 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ E We ∅)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ Ord ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4322 Tr wtr 5266 E cep 5581 We wwe 5632 Ord word 6370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-tr 5267 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6374 |
This theorem is referenced by: 0elon 6425 ord0eln0 6426 ordzsl 7850 smo0 8379 oicl 9554 alephgeom 10107 |
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