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Theorem frss 5599
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 3950 . . . . . 6 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 32 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32anim1d 611 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → (𝑥𝐵𝑥 ≠ ∅)))
43imim1d 82 . . 3 (𝐴𝐵 → (((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
54alimdv 1919 . 2 (𝐴𝐵 → (∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
6 df-fr 5587 . 2 (𝑅 Fr 𝐵 ↔ ∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
7 df-fr 5587 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
85, 6, 73imtr4g 295 1 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1539  wne 2942  wral 3063  wrex 3072  wss 3909  c0 4281   class class class wbr 5104   Fr wfr 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3446  df-in 3916  df-ss 3926  df-fr 5587
This theorem is referenced by:  freq2  5603  wess  5619  fprlem1  8228  frmin  9682  frrlem15  9690
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