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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 5278 | . 2 ⊢ Tr ∅ | |
2 | we0 5684 | . 2 ⊢ E We ∅ | |
3 | df-ord 6389 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ E We ∅)) | |
4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ Ord ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4339 Tr wtr 5265 E cep 5588 We wwe 5640 Ord word 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-tr 5266 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 |
This theorem is referenced by: 0elon 6440 ord0eln0 6441 ordzsl 7866 smo0 8397 oicl 9567 alephgeom 10120 pw2bday 28433 |
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