Theorem n0lts1e0 28633

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 28588	. . 3 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
2		0no 28074	. . 3 ⊢ 0s ∈ No
3		lestri3 27991	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 = 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
4	1, 2, 3	sylancl 598	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
5		n0sge0 28603	. . 3 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
6	5	biantrud 541	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
7		0n0s 28594	. . . 4 ⊢ 0s ∈ ℕ0s
8		n0lesltp1 28631	. . . 4 ⊢ ((𝐴 ∈ ℕ0s ∧ 0s ∈ ℕ0s) → (𝐴 ≤s 0s ↔ 𝐴 <s ( 0s +s 1s )))
9	7, 8	mpan2 704	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ 𝐴 <s ( 0s +s 1s )))
10		1no 28075	. . . . 5 ⊢ 1s ∈ No
11		addslid 28233	. . . . 5 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
12	10, 11	ax-mp 5	. . . 4 ⊢ ( 0s +s 1s ) = 1s
13	12	breq2i 5111	. . 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 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  (class class class)co 7413   No csur 27876   <s clts 27877   ≤s cles 27980   0_s c0s 28070   1_s c1s 28071   +_s cadds 28224  ℕ_0scn0s 28577

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27879  df-lts 27880  df-bday 27881  df-les 27981  df-slts 28023  df-cuts 28025  df-0s 28072  df-1s 28073  df-made 28092  df-old 28093  df-left 28095  df-right 28096  df-norec 28203  df-norec2 28214  df-adds 28225  df-negs 28286  df-subs 28287  df-n0s 28579  df-nns 28580

This theorem is used by:  bdaypw2n0bnd  28729  bdayfinbndlem1  28732