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Theorem frss 5634
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 3982 . . . . . 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 1911 . 2 (𝐴𝐵 → (∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
6 df-fr 5622 . 2 (𝑅 Fr 𝐵 ↔ ∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
7 df-fr 5622 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
85, 6, 73imtr4g 296 1 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1531  wne 2932  wral 3053  wrex 3062  wss 3941  c0 4315   class class class wbr 5139   Fr wfr 5619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-fr 5622
This theorem is referenced by:  freq2  5638  wess  5654  fprlem1  8281  frmin  9741  frrlem15  9749
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