Theorem n0lts1e0 28561

Description: A non-negative surreal integer is less than one iff it is zero. (Contributed by Scott Fenton, 23-Feb-2026.)

Assertion

Ref	Expression
n0lts1e0	⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))

Proof of Theorem n0lts1e0

Step	Hyp	Ref	Expression
1		n0no 28516	. . 3 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
2		0no 28002	. . 3 ⊢ 0s ∈ No
3		lestri3 27919	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 = 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
4	1, 2, 3	sylancl 597	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
5		n0sge0 28531	. . 3 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
6	5	biantrud 540	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
7		0n0s 28522	. . . 4 ⊢ 0s ∈ ℕ0s
8		n0lesltp1 28559	. . . 4 ⊢ ((𝐴 ∈ ℕ0s ∧ 0s ∈ ℕ0s) → (𝐴 ≤s 0s ↔ 𝐴 <s ( 0s +s 1s )))
9	7, 8	mpan2 703	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ 𝐴 <s ( 0s +s 1s )))
10		1no 28003	. . . . 5 ⊢ 1s ∈ No
11		addslid 28161	. . . . 5 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
12	10, 11	ax-mp 5	. . . 4 ⊢ ( 0s +s 1s ) = 1s
13	12	breq2i 5117	. . 3 ⊢ (𝐴 <s ( 0s +s 1s ) ↔ 𝐴 <s 1s )
14	9, 13	bitrdi 290	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ 𝐴 <s 1s ))
15	4, 6, 14	3bitr2rd 311	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   class class class wbr 5109  (class class class)co 7410   No csur 27804   <s clts 27805   ≤s cles 27908   0_s c0s 27998   1_s c1s 27999   +_s cadds 28152  ℕ_0scn0s 28505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-nadd 8648  df-no 27807  df-lts 27808  df-bday 27809  df-les 27909  df-slts 27951  df-cuts 27953  df-0s 28000  df-1s 28001  df-made 28020  df-old 28021  df-left 28023  df-right 28024  df-norec 28131  df-norec2 28142  df-adds 28153  df-negs 28214  df-subs 28215  df-n0s 28507  df-nns 28508

This theorem is referenced by:  bdaypw2n0bnd  28657  bdayfinbndlem1  28660