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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 5641 | . . 3 ⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | |
2 | sosn 5640 | . . 3 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | |
3 | 1, 2 | anbi12d 632 | . 2 ⊢ (Rel 𝑅 → ((𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}) ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴))) |
4 | df-we 5518 | . 2 ⊢ (𝑅 We {𝐴} ↔ (𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴})) | |
5 | pm4.24 566 | . 2 ⊢ (¬ 𝐴𝑅𝐴 ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴)) | |
6 | 3, 4, 5 | 3bitr4g 316 | 1 ⊢ (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 {csn 4569 class class class wbr 5068 Or wor 5475 Fr wfr 5513 We wwe 5515 Rel wrel 5562 |
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 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 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-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 |
This theorem is referenced by: 0we1 8133 canthwe 10075 |
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