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Theorem ssltsepc 27853
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 27850 . . 3 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
2 breq1 5151 . . . . 5 (𝑥 = 𝑋 → (𝑥 <s 𝑦𝑋 <s 𝑦))
3 breq2 5152 . . . . 5 (𝑦 = 𝑌 → (𝑋 <s 𝑦𝑋 <s 𝑌))
42, 3rspc2va 3634 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝑋 <s 𝑌)
54ancoms 458 . . 3 ((∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 <s 𝑌)
61, 5sylan 580 . 2 ((𝐴 <<s 𝐵 ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 <s 𝑌)
763impb 1114 1 ((𝐴 <<s 𝐵𝑋𝐴𝑌𝐵) → 𝑋 <s 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2106  wral 3059   class class class wbr 5148   <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:  ssltsepcd  27854  ssltun1  27868  ssltun2  27869  ssltdisj  27881  cofcutr  27973
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