Theorem n0lts1e0 28527

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 28482	. . 3 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
2		0no 27968	. . 3 ⊢ 0s ∈ No
3		lestri3 27885	. . 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 28497	. . 3 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
6	5	biantrud 540	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≤s 0s ↔ (𝐴 ≤s 0s ∧ 0s ≤s 𝐴)))
7		0n0s 28488	. . . 4 ⊢ 0s ∈ ℕ0s
8		n0lesltp1 28525	. . . 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 27969	. . . . 5 ⊢ 1s ∈ No
11		addslid 28127	. . . . 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 ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   class class class wbr 5111  (class class class)co 7411   No csur 27770   <s clts 27771   ≤s cles 27874   0_s c0s 27964   1_s c1s 27965   +_s cadds 28118  ℕ_0scn0s 28471

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 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

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 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-nadd 8652  df-no 27773  df-lts 27774  df-bday 27775  df-les 27875  df-slts 27917  df-cuts 27919  df-0s 27966  df-1s 27967  df-made 27986  df-old 27987  df-left 27989  df-right 27990  df-norec 28097  df-norec2 28108  df-adds 28119  df-negs 28180  df-subs 28181  df-n0s 28473  df-nns 28474

This theorem is referenced by:  bdaypw2n0bnd  28623  bdayfinbndlem1  28626