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| Description: Surreal less-than or equal is reflexive. Theorem 0(iii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| slerflex | ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sltirr 27792 | . 2 ⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴) | |
| 2 | slenlt 27798 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ∈ No ) → (𝐴 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐴)) | |
| 3 | 2 | anidms 566 | . 2 ⊢ (𝐴 ∈ No → (𝐴 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐴)) | 
| 4 | 1, 3 | mpbird 257 | 1 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∈ wcel 2107 class class class wbr 5142 No csur 27685 <s cslt 27686 ≤s csle 27790 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-1o 8507 df-2o 8508 df-no 27688 df-slt 27689 df-sle 27791 | 
| This theorem is referenced by: maxs1 27811 maxs2 27812 mins1 27813 mins2 27814 0slt1s 27875 cofcutrtime 27962 cofss 27965 coiniss 27966 cutlt 27967 cutmax 27969 cutmin 27970 slemuld 28165 mulsge0d 28173 slemul1ad 28209 abs0s 28267 sleabs 28273 n0sge0 28342 uzsind 28392 zscut 28394 nohalf 28408 halfcut 28417 addhalfcut 28420 | 
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