Theorem sess1 5082

Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
sess1	|- ( R C_ S -> ( S Se A -> R Se A ) )

Proof of Theorem sess1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( R C_ S /\ y e. A ) -> R C_ S )
2	1	ssbrd 4696	. . . . 5 |- ( ( R C_ S /\ y e. A ) -> ( y R x -> y S x ) )
3	2	ss2rabdv 3683	. . . 4 |- ( R C_ S -> { y e. A | y R x } C_ { y e. A | y S x } )
4		ssexg 4804	. . . . 5 |- ( ( { y e. A | y R x } C_ { y e. A | y S x } /\ { y e. A | y S x } e. _V ) -> { y e. A | y R x } e. _V )
5	4	ex 450	. . . 4 |- ( { y e. A | y R x } C_ { y e. A | y S x } -> ( { y e. A | y S x } e. _V -> { y e. A | y R x } e. _V ) )
6	3, 5	syl 17	. . 3 |- ( R C_ S -> ( { y e. A | y S x } e. _V -> { y e. A | y R x } e. _V ) )
7	6	ralimdv 2963	. 2 |- ( R C_ S -> ( A. x e. A { y e. A | y S x } e. _V -> A. x e. A { y e. A | y R x } e. _V ) )
8		df-se 5074	. 2 |- ( S Se A <-> A. x e. A { y e. A | y S x } e. _V )
9		df-se 5074	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
10	7, 8, 9	3imtr4g 285	1 |- ( R C_ S -> ( S Se A -> R Se A ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Se wse 5071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-br 4654  df-se 5074

This theorem is referenced by:  seeq1  5086