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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 5498 | . 2 ⊢ 𝑅 Fr ∅ | |
2 | so0 5473 | . 2 ⊢ 𝑅 Or ∅ | |
3 | df-we 5480 | . 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅)) | |
4 | 1, 2, 3 | mpbir2an 710 | 1 ⊢ 𝑅 We ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4243 Or wor 5437 Fr wfr 5475 We wwe 5477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 |
This theorem is referenced by: ord0 6211 cantnf0 9122 cantnf 9140 wemapwe 9144 ltweuz 13324 |
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