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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 5176 | . 2 ⊢ Tr ∅ | |
2 | we0 5543 | . 2 ⊢ E We ∅ | |
3 | df-ord 6187 | . 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ E We ∅)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ Ord ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4284 Tr wtr 5165 E cep 5457 We wwe 5506 Ord word 6183 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-in 3936 df-ss 3945 df-nul 4285 df-pw 4534 df-uni 4832 df-tr 5166 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 |
This theorem is referenced by: 0elon 6237 ord0eln0 6238 ordzsl 7553 smo0 7988 oicl 8986 alephgeom 9501 |
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