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

Theorem ssltsepc 27154
Description: Two elements of separated sets obey less-than. (Contributed by Scott Fenton, 20-Aug-2024.)
Assertion
Ref Expression
ssltsepc ((𝐴 <<s 𝐵𝑋𝐴𝑌𝐵) → 𝑋 <s 𝑌)

Proof of Theorem ssltsepc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltsep 27152 . . 3 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
2 breq1 5109 . . . . 5 (𝑥 = 𝑋 → (𝑥 <s 𝑦𝑋 <s 𝑦))
3 breq2 5110 . . . . 5 (𝑦 = 𝑌 → (𝑋 <s 𝑦𝑋 <s 𝑌))
42, 3rspc2va 3590 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝑋 <s 𝑌)
54ancoms 460 . . 3 ((∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 <s 𝑌)
61, 5sylan 581 . 2 ((𝐴 <<s 𝐵 ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 <s 𝑌)
763impb 1116 1 ((𝐴 <<s 𝐵𝑋𝐴𝑌𝐵) → 𝑋 <s 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wral 3061   class class class wbr 5106   <s cslt 27005   <<s csslt 27142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-sslt 27143
This theorem is referenced by:  ssltsepcd  27155  ssltun1  27169  ssltun2  27170  ssltdisj  27182  cofcutr  27265
  Copyright terms: Public domain W3C validator