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Theorem ssltsepc 33642
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 33640 . . 3 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
2 breq1 5043 . . . . 5 (𝑥 = 𝑋 → (𝑥 <s 𝑦𝑋 <s 𝑦))
3 breq2 5044 . . . . 5 (𝑦 = 𝑌 → (𝑋 <s 𝑦𝑋 <s 𝑌))
42, 3rspc2va 3540 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝑋 <s 𝑌)
54ancoms 462 . . 3 ((∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 <s 𝑌)
61, 5sylan 583 . 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 399  w3a 1088  wcel 2114  wral 3054   class class class wbr 5040   <s cslt 33499   <<s csslt 33630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-xp 5541  df-sslt 33631
This theorem is referenced by:  ssltsepcd  33643  ssltun1  33657  ssltun2  33658  ssltdisj  33670  cofcutr  33747
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