Theorem ssltsn 27761

Description: Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.)

Hypotheses

Ref	Expression
ssltsn.1	⊢ (𝜑 → 𝐴 ∈ No )
ssltsn.2	⊢ (𝜑 → 𝐵 ∈ No )
ssltsn.3	⊢ (𝜑 → 𝐴 <s 𝐵)

Assertion

Ref	Expression
ssltsn	⊢ (𝜑 → {𝐴} <<s {𝐵})

Proof of Theorem ssltsn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5411	. . 3 ⊢ {𝐴} ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝐴} ∈ V)
3		snex 5411	. . 3 ⊢ {𝐵} ∈ V
4	3	a1i 11	. 2 ⊢ (𝜑 → {𝐵} ∈ V)
5		ssltsn.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
6	5	snssd 4790	. 2 ⊢ (𝜑 → {𝐴} ⊆ No )
7		ssltsn.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
8	7	snssd 4790	. 2 ⊢ (𝜑 → {𝐵} ⊆ No )
9		ssltsn.3	. . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵)
10		velsn 4622	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		velsn 4622	. . . . 5 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
12		breq12 5129	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝐵))
13	10, 11, 12	syl2anb 598	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝐵))
14	9, 13	syl5ibrcom 247	. . 3 ⊢ (𝜑 → ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦))
15	14	3impib 1116	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦)
16	2, 4, 6, 8, 15	ssltd 27760	1 ⊢ (𝜑 → {𝐴} <<s {𝐵})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464  {csn 4606   class class class wbr 5124   No csur 27608   <s cslt 27609   <<s csslt 27749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-sslt 27750

This theorem is referenced by:  cutneg  27802  zscut  28352  twocut  28366  nohalf  28367  pw2recs  28380  halfcut  28390  addhalfcut  28391  zs12bday  28400