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Theorem frss 4991
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 3570 . . . . . 6 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 32 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32anim1d 585 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → (𝑥𝐵𝑥 ≠ ∅)))
43imim1d 79 . . 3 (𝐴𝐵 → (((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
54alimdv 1830 . 2 (𝐴𝐵 → (∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)))
6 df-fr 4983 . 2 (𝑅 Fr 𝐵 ↔ ∀𝑥((𝑥𝐵𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
7 df-fr 4983 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
85, 6, 73imtr4g 283 1 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1472  wne 2775  wral 2891  wrex 2892  wss 3535  c0 3869   class class class wbr 4573   Fr wfr 4980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-in 3542  df-ss 3549  df-fr 4983
This theorem is referenced by:  freq2  4995  wess  5011  frmin  30785  frrlem5  30830
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