Theorem ssltsn 27770

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 27769	. 2 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → {𝐴} <<s {𝐵})

Syntax hints:   → wi 4   ∈ wcel 2114  {csn 4581   class class class wbr 5099   No csur 27611   <s cslt 27612   <<s csslt 27757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-sslt 27758

This theorem is referenced by:  cutneg  27814  zscut  28386  twocut  28402  nohalf  28403  pw2recs  28417  halfcut  28437  addhalfcut  28438  pw2cut2  28441  bdaypw2n0sbndlem  28442  bdayfinbndlem1  28446