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Mirrors > Home > MPE Home > Th. List > Mathboxes > slerflex | Structured version Visualization version GIF version |
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 33720 | . 2 ⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴) | |
2 | slenlt 33726 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ∈ No ) → (𝐴 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐴)) | |
3 | 2 | anidms 570 | . 2 ⊢ (𝐴 ∈ No → (𝐴 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐴)) |
4 | 1, 3 | mpbird 260 | 1 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∈ wcel 2112 class class class wbr 5070 No csur 33614 <s cslt 33615 ≤s csle 33718 |
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-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pr 5339 ax-un 7545 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5153 df-tr 5179 df-id 5472 df-eprel 5478 df-po 5486 df-so 5487 df-fr 5527 df-we 5529 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-ord 6237 df-on 6238 df-suc 6240 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-fv 6409 df-1o 8226 df-2o 8227 df-no 33617 df-slt 33618 df-sle 33719 |
This theorem is referenced by: 0slt1s 33794 cofcutrtime 33864 |
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