Theorem sess1 4197

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 108	. . . . . 6 |- ( ( R C_ S /\ y e. A ) -> R C_ S )
2	1	ssbrd 3916	. . . . 5 |- ( ( R C_ S /\ y e. A ) -> ( y R x -> y S x ) )
3	2	ss2rabdv 3125	. . . 4 |- ( R C_ S -> { y e. A | y R x } C_ { y e. A | y S x } )
4		ssexg 4007	. . . . 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 114	. . . 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 14	. . 3 |- ( R C_ S -> ( { y e. A | y S x } e. _V -> { y e. A | y R x } e. _V ) )
7	6	ralimdv 2459	. 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 4193	. 2 |- ( S Se A <-> A. x e. A { y e. A | y S x } e. _V )
9		df-se 4193	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
10	7, 8, 9	3imtr4g 204	1 |- ( R C_ S -> ( S Se A -> R Se A ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1448   A.wral 2375   {crab 2379   _Vcvv 2641   C_ wss 3021   class class class wbr 3875   Se wse 4189

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rab 2384  df-v 2643  df-in 3027  df-ss 3034  df-br 3876  df-se 4193

This theorem is referenced by:  seeq1  4199