Theorem n0lts1e0 28592

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 28547	. . 3 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
2		0no 28033	. . 3 ⊢ 0s ∈ No
3		lestri3 27950	. . 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 28562	. . 3 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
6	5	biantrud 541	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
7		0n0s 28553	. . . 4 ⊢ 0s ∈ ℕ0s
8		n0lesltp1 28590	. . . 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 28034	. . . . 5 ⊢ 1s ∈ No
11		addslid 28192	. . . . 5 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
12	10, 11	ax-mp 5	. . . 4 ⊢ ( 0s +s 1s ) = 1s
13	12	breq2i 5119	. . 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 2146   class class class wbr 5111  (class class class)co 7416   No csur 27835   <s clts 27836   ≤s cles 27939   0_s c0s 28029   1_s c1s 28030   +_s cadds 28183  ℕ_0scn0s 28536

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27838  df-lts 27839  df-bday 27840  df-les 27940  df-slts 27982  df-cuts 27984  df-0s 28031  df-1s 28032  df-made 28051  df-old 28052  df-left 28054  df-right 28055  df-norec 28162  df-norec2 28173  df-adds 28184  df-negs 28245  df-subs 28246  df-n0s 28538  df-nns 28539

This theorem is used by:  bdaypw2n0bnd  28688  bdayfinbndlem1  28691