Theorem ssltsn 27726

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

Step	Hyp	Ref	Expression
1		ssltsn.3	. 2 ⊢ (𝜑 → 𝐴 <s 𝐵)
2		ssltsn.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		ssltsn.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
4	2, 3	ssltsnb 27725	. 2 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → {𝐴} <<s {𝐵})

Syntax hints:   → wi 4   ∈ wcel 2110  {csn 4574   class class class wbr 5089   No csur 27571   <s cslt 27572   <<s csslt 27713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-xp 5620  df-sslt 27714

This theorem is referenced by:  cutneg  27770  zscut  28324  twocut  28339  nohalf  28340  pw2recs  28354  halfcut  28371  addhalfcut  28372  pw2cut2  28375  zs12bday  28387