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Mirrors > Home > MPE Home > Th. List > wesn | Structured version Visualization version GIF version |
Description: Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
wesn | ⊢ (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frsn 5763 | . . 3 ⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | |
2 | sosn 5762 | . . 3 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | |
3 | 1, 2 | anbi12d 631 | . 2 ⊢ (Rel 𝑅 → ((𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}) ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴))) |
4 | df-we 5633 | . 2 ⊢ (𝑅 We {𝐴} ↔ (𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴})) | |
5 | pm4.24 564 | . 2 ⊢ (¬ 𝐴𝑅𝐴 ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴)) | |
6 | 3, 4, 5 | 3bitr4g 313 | 1 ⊢ (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 {csn 4628 class class class wbr 5148 Or wor 5587 Fr wfr 5628 We wwe 5630 Rel wrel 5681 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 |
This theorem is referenced by: 0we1 8505 canthwe 10645 |
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