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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 5651 | . 2 ⊢ 𝑅 Fr ∅ | |
2 | so0 5620 | . 2 ⊢ 𝑅 Or ∅ | |
3 | df-we 5629 | . 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 4318 Or wor 5583 Fr wfr 5624 We wwe 5626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 |
This theorem is referenced by: ord0 6416 cantnf0 9690 cantnf 9708 wemapwe 9712 ltweuz 13950 |
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