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Theorem ssltsepc 27531
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 27528 . . 3 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
2 breq1 5150 . . . . 5 (𝑥 = 𝑋 → (𝑥 <s 𝑦𝑋 <s 𝑦))
3 breq2 5151 . . . . 5 (𝑦 = 𝑌 → (𝑋 <s 𝑦𝑋 <s 𝑌))
42, 3rspc2va 3622 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝑋 <s 𝑌)
54ancoms 457 . . 3 ((∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 <s 𝑌)
61, 5sylan 578 . 2 ((𝐴 <<s 𝐵 ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 <s 𝑌)
763impb 1113 1 ((𝐴 <<s 𝐵𝑋𝐴𝑌𝐵) → 𝑋 <s 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085  wcel 2104  wral 3059   class class class wbr 5147   <s cslt 27380   <<s csslt 27518
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-sslt 27519
This theorem is referenced by:  ssltsepcd  27532  ssltun1  27546  ssltun2  27547  ssltdisj  27559  cofcutr  27649
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