Theorem relss 3236

Description: Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.

Assertion

Ref	Expression
relss	|- (A (_ B -> (Rel B -> Rel A))

Proof of Theorem relss

Step	Hyp	Ref	Expression
1		sstr2 2061	. 2 |- (A (_ B -> (B (_ (V X. V) -> A (_ (V X. V)))
2		df-rel 3175	. 2 |- (Rel B <-> B (_ (V X. V))
3		df-rel 3175	. 2 |- (Rel A <-> A (_ (V X. V))
4	1, 2, 3	3imtr4g 551	1 |- (A (_ B -> (Rel B -> Rel A))

Syntax hints:   -> wi 3  Vcvv 1802   (_ wss 2037   X. cxp 3158  Rel wrel 3165

This theorem is referenced by:  relin1 3252  relin2 3253  reldif 3254  iss 3381  intasym 3422  asymref 3423  asymrefOLD 3425  intirr 3427  funss 3520  funssres 3538  prcdpq 5069  phrel 8405  bnrel 8458  hlrel 8525

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-in 2041  df-ss 2043  df-rel 3175

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