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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 5548 | . 2 ⊢ 𝑅 Fr ∅ | |
2 | so0 5522 | . 2 ⊢ 𝑅 Or ∅ | |
3 | df-we 5529 | . 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅)) | |
4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ 𝑅 We ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4254 Or wor 5485 Fr wfr 5524 We wwe 5526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-po 5486 df-so 5487 df-fr 5527 df-we 5529 |
This theorem is referenced by: ord0 6286 cantnf0 9320 cantnf 9338 wemapwe 9342 ltweuz 13566 |
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