![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > we0 | Structured version Visualization version GIF version |
Description: Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.) |
Ref | Expression |
---|---|
we0 | ⊢ 𝑅 We ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fr0 5322 | . 2 ⊢ 𝑅 Fr ∅ | |
2 | so0 5297 | . 2 ⊢ 𝑅 Or ∅ | |
3 | df-we 5304 | . 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅)) | |
4 | 1, 2, 3 | mpbir2an 704 | 1 ⊢ 𝑅 We ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4145 Or wor 5263 Fr wfr 5299 We wwe 5301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-in 3806 df-ss 3813 df-nul 4146 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 |
This theorem is referenced by: ord0 6016 cantnf0 8850 cantnf 8868 wemapwe 8872 ltweuz 13056 |
Copyright terms: Public domain | W3C validator |