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Theorem ssltsn 27837
Description: Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.)
Hypotheses
Ref Expression
ssltsn.1 (𝜑𝐴 No )
ssltsn.2 (𝜑𝐵 No )
ssltsn.3 (𝜑𝐴 <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.
StepHypRef Expression
1 snex 5436 . . 3 {𝐴} ∈ V
21a1i 11 . 2 (𝜑 → {𝐴} ∈ V)
3 snex 5436 . . 3 {𝐵} ∈ V
43a1i 11 . 2 (𝜑 → {𝐵} ∈ V)
5 ssltsn.1 . . 3 (𝜑𝐴 No )
65snssd 4809 . 2 (𝜑 → {𝐴} ⊆ No )
7 ssltsn.2 . . 3 (𝜑𝐵 No )
87snssd 4809 . 2 (𝜑 → {𝐵} ⊆ No )
9 ssltsn.3 . . . 4 (𝜑𝐴 <s 𝐵)
10 velsn 4642 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11 velsn 4642 . . . . 5 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
12 breq12 5148 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 <s 𝑦𝐴 <s 𝐵))
1310, 11, 12syl2anb 598 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → (𝑥 <s 𝑦𝐴 <s 𝐵))
149, 13syl5ibrcom 247 . . 3 (𝜑 → ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦))
15143impib 1117 . 2 ((𝜑𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦)
162, 4, 6, 8, 15ssltd 27836 1 (𝜑 → {𝐴} <<s {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626   class class class wbr 5143   No csur 27684   <s cslt 27685   <<s csslt 27825
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-sslt 27826
This theorem is referenced by:  n0scut  28338  zscut  28393  halfcut  28416  pw2bday  28418  addhalfcut  28419  zs12bday  28424
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