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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. (Contributed by NM, 11-May-1994.) |
Ref | Expression |
---|---|
ord0 | ⊢ Ord ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tr0 5068 | . 2 ⊢ Tr ∅ | |
2 | we0 5430 | . 2 ⊢ E We ∅ | |
3 | df-ord 6061 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ E We ∅)) | |
4 | 1, 2, 3 | mpbir2an 707 | 1 ⊢ Ord ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4206 Tr wtr 5057 E cep 5344 We wwe 5393 Ord word 6057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-dif 3857 df-in 3861 df-ss 3869 df-nul 4207 df-pw 4449 df-uni 4740 df-tr 5058 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-ord 6061 |
This theorem is referenced by: 0elon 6111 ord0eln0 6112 ordzsl 7407 smo0 7838 oicl 8829 alephgeom 9343 |
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