Theorem ssltsn 27739

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 27738	. 2 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → {𝐴} <<s {𝐵})

Syntax hints:   → wi 4   ∈ wcel 2111  {csn 4575   class class class wbr 5093   No csur 27584   <s cslt 27585   <<s csslt 27726

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-sslt 27727

This theorem is referenced by:  cutneg  27783  zscut  28337  twocut  28352  nohalf  28353  pw2recs  28367  halfcut  28384  addhalfcut  28385  pw2cut2  28388  zs12bday  28400