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| Mirrors > Home > MPE Home > Th. List > scutcld | Structured version Visualization version GIF version | ||
| Description: Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| scutcld.1 | ⊢ (𝜑 → 𝐴 <<s 𝐵) | 
| Ref | Expression | 
|---|---|
| scutcld | ⊢ (𝜑 → (𝐴 |s 𝐵) ∈ No ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | scutcld.1 | . 2 ⊢ (𝜑 → 𝐴 <<s 𝐵) | |
| 2 | scutcl 27847 | . 2 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No ) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 |s 𝐵) ∈ No ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 No csur 27684 <<s csslt 27825 |s cscut 27827 | 
| 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-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-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-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 | 
| This theorem is referenced by: cofcut1 27954 cofcutr 27958 addsuniflem 28034 negsunif 28087 ssltmul1 28173 ssltmul2 28174 mulsuniflem 28175 mulsunif2lem 28195 precsexlem11 28241 precsex 28242 elons2 28281 zscut 28393 nohalf 28407 halfcut 28416 | 
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