Theorem relss 3246

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 2071	. 2 |- (A (_ B -> (B (_ (V X. V) -> A (_ (V X. V)))
2		df-rel 3185	. 2 |- (Rel B <-> B (_ (V X. V))
3		df-rel 3185	. 2 |- (Rel A <-> A (_ (V X. V))
4	1, 2, 3	3imtr4g 553	1 |- (A (_ B -> (Rel B -> Rel A))

Syntax hints:   -> wi 3  Vcvv 1811   (_ wss 2047   X. cxp 3168  Rel wrel 3175

This theorem is referenced by:  relin1 3262  relin2 3263  reldif 3264  iss 3397  intasym 3438  asymref 3439  intirr 3441  funss 3534  funssres 3552  prcdpq 5097  phrel 8474  bnrel 8527  hlrel 8594

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-rel 3185

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