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Theorem frss 5515
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 3971 . . . . . 6 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 32 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32anim1d 610 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → (𝑥𝐵𝑥 ≠ ∅)))
43imim1d 82 . . 3 (𝐴𝐵 → (((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
54alimdv 1908 . 2 (𝐴𝐵 → (∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
6 df-fr 5507 . 2 (𝑅 Fr 𝐵 ↔ ∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
7 df-fr 5507 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
85, 6, 73imtr4g 297 1 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526  wne 3013  wral 3135  wrex 3136  wss 3933  c0 4288   class class class wbr 5057   Fr wfr 5504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-in 3940  df-ss 3949  df-fr 5507
This theorem is referenced by:  freq2  5519  wess  5535  frmin  32981  fprlem1  33034  frrlem15  33039
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