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Theorem frss 5051
Description: Subset theorem for the well-founded predicate. Exercise 1 of [TakeutiZaring] p. 31. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
frss (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))

Proof of Theorem frss
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr2 3595 . . . . . 6 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 32 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32anim1d 587 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → (𝑥𝐵𝑥 ≠ ∅)))
43imim1d 82 . . 3 (𝐴𝐵 → (((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
54alimdv 1842 . 2 (𝐴𝐵 → (∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
6 df-fr 5043 . 2 (𝑅 Fr 𝐵 ↔ ∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
7 df-fr 5043 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
85, 6, 73imtr4g 285 1 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478  wne 2790  wral 2908  wrex 2909  wss 3560  c0 3897   class class class wbr 4623   Fr wfr 5040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3567  df-ss 3574  df-fr 5043
This theorem is referenced by:  freq2  5055  wess  5071  frmin  31493  frrlem5  31538
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