Theorem ssltsn 27738

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 5433	. . 3 ⊢ {𝐴} ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝐴} ∈ V)
3		snex 5433	. . 3 ⊢ {𝐵} ∈ V
4	3	a1i 11	. 2 ⊢ (𝜑 → {𝐵} ∈ V)
5		ssltsn.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
6	5	snssd 4813	. 2 ⊢ (𝜑 → {𝐴} ⊆ No )
7		ssltsn.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
8	7	snssd 4813	. 2 ⊢ (𝜑 → {𝐵} ⊆ No )
9		ssltsn.3	. . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵)
10		velsn 4645	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11		velsn 4645	. . . . 5 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
12		breq12 5153	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝐵))
13	10, 11, 12	syl2anb 597	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝐵))
14	9, 13	syl5ibrcom 246	. . 3 ⊢ (𝜑 → ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦))
15	14	3impib 1114	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦)
16	2, 4, 6, 8, 15	ssltd 27737	1 ⊢ (𝜑 → {𝐴} <<s {𝐵})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471  {csn 4629   class class class wbr 5148   No csur 27586   <s cslt 27587   <<s csslt 27726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-sslt 27727

This theorem is referenced by:  n0scut  28216