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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 5536 | . 2 ⊢ 𝑅 Fr ∅ | |
2 | so0 5511 | . 2 ⊢ 𝑅 Or ∅ | |
3 | df-we 5518 | . 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ 𝑅 We ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4293 Or wor 5475 Fr wfr 5513 We wwe 5515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 |
This theorem is referenced by: ord0 6245 cantnf0 9140 cantnf 9158 wemapwe 9162 ltweuz 13332 |
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