Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj93 Structured version   Visualization version   GIF version

Theorem bnj93 31261
 Description: Technical lemma for bnj97 31264. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj93 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem bnj93
StepHypRef Expression
1 df-bnj15 31089 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
21simprbi 483 . . 3 (𝑅 FrSe 𝐴𝑅 Se 𝐴)
3 df-bnj13 31087 . . 3 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
42, 3sylib 208 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
54r19.21bi 3070 1 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   Fr wfr 5222   predc-bnj14 31084   Se w-bnj13 31086   FrSe w-bnj15 31088 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-ral 3055  df-bnj13 31087  df-bnj15 31089 This theorem is referenced by:  bnj96  31263  bnj97  31264  bnj149  31273  bnj150  31274  bnj518  31284  bnj1148  31392
 Copyright terms: Public domain W3C validator