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| Mirrors > Home > MPE Home > Th. List > lltropt | Structured version Visualization version GIF version | ||
| Description: The left options of a surreal are strictly less than the right options of the same surreal. (Contributed by Scott Fenton, 6-Aug-2024.) (Revised by Scott Fenton, 21-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| lltropt | ⊢ ( L ‘𝐴) <<s ( R ‘𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssltleft 27909 | . . 3 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴}) | |
| 2 | ssltright 27910 | . . 3 ⊢ (𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴)) | |
| 3 | snnzg 4774 | . . 3 ⊢ (𝐴 ∈ No → {𝐴} ≠ ∅) | |
| 4 | sslttr 27852 | . . 3 ⊢ ((( L ‘𝐴) <<s {𝐴} ∧ {𝐴} <<s ( R ‘𝐴) ∧ {𝐴} ≠ ∅) → ( L ‘𝐴) <<s ( R ‘𝐴)) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s ( R ‘𝐴)) | 
| 6 | 0elpw 5356 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
| 7 | nulssgt 27843 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
| 8 | 6, 7 | mp1i 13 | . . 3 ⊢ (¬ 𝐴 ∈ No → ∅ <<s ∅) | 
| 9 | leftf 27904 | . . . . . 6 ⊢ L : No ⟶𝒫 No | |
| 10 | 9 | fdmi 6747 | . . . . 5 ⊢ dom L = No | 
| 11 | 10 | eleq2i 2833 | . . . 4 ⊢ (𝐴 ∈ dom L ↔ 𝐴 ∈ No ) | 
| 12 | ndmfv 6941 | . . . 4 ⊢ (¬ 𝐴 ∈ dom L → ( L ‘𝐴) = ∅) | |
| 13 | 11, 12 | sylnbir 331 | . . 3 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅) | 
| 14 | rightf 27905 | . . . . . 6 ⊢ R : No ⟶𝒫 No | |
| 15 | 14 | fdmi 6747 | . . . . 5 ⊢ dom R = No | 
| 16 | 15 | eleq2i 2833 | . . . 4 ⊢ (𝐴 ∈ dom R ↔ 𝐴 ∈ No ) | 
| 17 | ndmfv 6941 | . . . 4 ⊢ (¬ 𝐴 ∈ dom R → ( R ‘𝐴) = ∅) | |
| 18 | 16, 17 | sylnbir 331 | . . 3 ⊢ (¬ 𝐴 ∈ No → ( R ‘𝐴) = ∅) | 
| 19 | 8, 13, 18 | 3brtr4d 5175 | . 2 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) <<s ( R ‘𝐴)) | 
| 20 | 5, 19 | pm2.61i 182 | 1 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 No csur 27684 <<s csslt 27825 L cleft 27884 R cright 27885 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-made 27886 df-old 27887 df-left 27889 df-right 27890 | 
| This theorem is referenced by: madebdaylemlrcut 27937 madebday 27938 scutfo 27942 sltn0 27943 sltlpss 27945 slelss 27946 cutpos 27967 addsproplem2 28003 addsasslem1 28036 addsasslem2 28037 negsproplem2 28061 negsid 28073 mulsproplem5 28146 mulsproplem6 28147 mulsproplem7 28148 mulsproplem8 28149 addsdilem1 28177 mulsasslem1 28189 mulsasslem2 28190 precsexlem11 28241 halfcut 28416 | 
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