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Theorem elnns2 28202

Description: A positive surreal integer is a non-negative surreal integer greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)

**Assertion**
Ref	Expression
elnns2	⊢ (𝐴 ∈ ℕ_s ↔ (𝐴 ∈ ℕ_0s ∧ 0_s <s 𝐴))

**Proof of Theorem elnns2**
Step	Hyp	Ref	Expression
1		elnns 28201	. 2 ⊢ (𝐴 ∈ ℕ_s ↔ (𝐴 ∈ ℕ_0s ∧ 𝐴 ≠ 0_s ))
2		nesym 2993	. . . . 5 ⊢ (𝐴 ≠ 0_s ↔ ¬ 0_s = 𝐴)
3		n0sge0 28199	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ_0s → 0_s ≤s 𝐴)
4		0sno 27752	. . . . . . . . 9 ⊢ 0_s ∈ No
5		n0sno 28188	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ_0s → 𝐴 ∈ No )
6		sleloe 27680	. . . . . . . . 9 ⊢ (( 0_s ∈ No ∧ 𝐴 ∈ No ) → ( 0_s ≤s 𝐴 ↔ ( 0_s <s 𝐴 ∨ 0_s = 𝐴)))
7	4, 5, 6	sylancr 586	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ_0s → ( 0_s ≤s 𝐴 ↔ ( 0_s <s 𝐴 ∨ 0_s = 𝐴)))
8	3, 7	mpbid 231	. . . . . . 7 ⊢ (𝐴 ∈ ℕ_0s → ( 0_s <s 𝐴 ∨ 0_s = 𝐴))
9	8	orcomd 870	. . . . . 6 ⊢ (𝐴 ∈ ℕ_0s → ( 0_s = 𝐴 ∨ 0_s <s 𝐴))
10	9	ord 863	. . . . 5 ⊢ (𝐴 ∈ ℕ_0s → (¬ 0_s = 𝐴 → 0_s <s 𝐴))
11	2, 10	biimtrid 241	. . . 4 ⊢ (𝐴 ∈ ℕ_0s → (𝐴 ≠ 0_s → 0_s <s 𝐴))
12		sgt0ne0 27760	. . . 4 ⊢ ( 0_s <s 𝐴 → 𝐴 ≠ 0_s )
13	11, 12	impbid1 224	. . 3 ⊢ (𝐴 ∈ ℕ_0s → (𝐴 ≠ 0_s ↔ 0_s <s 𝐴))
14	13	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ ℕ_0s ∧ 𝐴 ≠ 0_s ) ↔ (𝐴 ∈ ℕ_0s ∧ 0_s <s 𝐴))
15	1, 14	bitri 275	1 ⊢ (𝐴 ∈ ℕ_s ↔ (𝐴 ∈ ℕ_0s ∧ 0_s <s 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 No csur 27566 <s cslt 27567 ≤s csle 27670 0_s c0s 27748 ℕ_0scnn0s 28178 ℕ_scnns 28179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-nadd 8680 df-no 27569 df-slt 27570 df-bday 27571 df-sle 27671 df-sslt 27707 df-scut 27709 df-0s 27750 df-1s 27751 df-made 27767 df-old 27768 df-left 27770 df-right 27771 df-norec2 27859 df-adds 27870 df-n0s 28180 df-nns 28181

This theorem is referenced by: nnaddscl 28205 nnmulscl 28206

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