Theorem n0lts1e0 28523

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 28478	. . 3 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
2		0no 27964	. . 3 ⊢ 0s ∈ No
3		lestri3 27881	. . 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 28493	. . 3 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
6	5	biantrud 540	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
7		0n0s 28484	. . . 4 ⊢ 0s ∈ ℕ0s
8		n0lesltp1 28521	. . . 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 27965	. . . . 5 ⊢ 1s ∈ No
11		addslid 28123	. . . . 5 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
12	10, 11	ax-mp 5	. . . 4 ⊢ ( 0s +s 1s ) = 1s
13	12	breq2i 5118	. . 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 1567   ∈ wcel 2149   class class class wbr 5110  (class class class)co 7408   No csur 27766   <s clts 27767   ≤s cles 27870   0_s c0s 27960   1_s c1s 27961   +_s cadds 28114  ℕ_0scn0s 28467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  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 27769  df-lts 27770  df-bday 27771  df-les 27871  df-slts 27913  df-cuts 27915  df-0s 27962  df-1s 27963  df-made 27982  df-old 27983  df-left 27985  df-right 27986  df-norec 28093  df-norec2 28104  df-adds 28115  df-negs 28176  df-subs 28177  df-n0s 28469  df-nns 28470

This theorem is referenced by:  bdaypw2n0bnd  28619  bdayfinbndlem1  28622