MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssltsn Structured version   Visualization version   GIF version

Theorem ssltsn 27852
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 5442 . . 3 {𝐴} ∈ V
21a1i 11 . 2 (𝜑 → {𝐴} ∈ V)
3 snex 5442 . . 3 {𝐵} ∈ V
43a1i 11 . 2 (𝜑 → {𝐵} ∈ V)
5 ssltsn.1 . . 3 (𝜑𝐴 No )
65snssd 4814 . 2 (𝜑 → {𝐴} ⊆ No )
7 ssltsn.2 . . 3 (𝜑𝐵 No )
87snssd 4814 . 2 (𝜑 → {𝐵} ⊆ No )
9 ssltsn.3 . . . 4 (𝜑𝐴 <s 𝐵)
10 velsn 4647 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
11 velsn 4647 . . . . 5 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
12 breq12 5153 . . . . 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 1115 . 2 ((𝜑𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦)
162, 4, 6, 8, 15ssltd 27851 1 (𝜑 → {𝐴} <<s {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   class class class wbr 5148   No csur 27699   <s cslt 27700   <<s csslt 27840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-sslt 27841
This theorem is referenced by:  n0scut  28353  zscut  28408  halfcut  28431  pw2bday  28433  addhalfcut  28434  zs12bday  28439
  Copyright terms: Public domain W3C validator