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| Mirrors > Home > MPE Home > Th. List > nnsno | Structured version Visualization version GIF version | ||
| Description: A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.) |
| Ref | Expression |
|---|---|
| nnsno | ⊢ (𝐴 ∈ ℕs → 𝐴 ∈ No ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nnssno 28327 | . 2 ⊢ ℕs ⊆ No | |
| 2 | 1 | sseli 3979 | 1 ⊢ (𝐴 ∈ ℕs → 𝐴 ∈ No ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 No csur 27684 ℕscnns 28319 |
| 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-se 5638 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-lim 6389 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-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-0s 27869 df-1s 27870 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec2 27982 df-adds 27993 df-n0s 28320 df-nns 28321 |
| This theorem is referenced by: nnsnod 28331 nnsgt0 28342 nnzs 28372 znegscl 28378 zmulscld 28383 elzn0s 28384 eln0zs 28386 elnnzs 28387 2sno 28403 recut 28428 0reno 28429 renegscl 28430 readdscl 28431 remulscllem1 28432 remulscllem2 28433 remulscl 28434 |
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