Theorem ssrel 2479

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

Assertion

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

Proof of Theorem ssrel

Step	Hyp	Ref	Expression
1		sstr2 1510	. 2 |- (A (_ B -> (B (_ (V X. V) -> A (_ (V X. V)))
2		df-rel 2425	. 2 |- (Rel B <-> B (_ (V X. V))
3		df-rel 2425	. 2 |- (Rel A <-> A (_ (V X. V))
4	1, 2, 3	3imtr4g 426	1 |- (A (_ B -> (Rel B -> Rel A))

Syntax hints:   -> wi 2  Vcvv 1348   (_ wss 1487   X. cxp 2408  Rel wrel 2415

This theorem is referenced by:  relin 2491  reldif 2492  iss 2599  intasym 2627  intirr 2628  funss 2682  funssres 2698  prcdpq 3891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-rel 2425

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