Theorem sess1 4428

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 109	. . . . . 6 |- ( ( R C_ S /\ y e. A ) -> R C_ S )
2	1	ssbrd 4126	. . . . 5 |- ( ( R C_ S /\ y e. A ) -> ( y R x -> y S x ) )
3	2	ss2rabdv 3305	. . . 4 |- ( R C_ S -> { y e. A | y R x } C_ { y e. A | y S x } )
4		ssexg 4223	. . . . 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 115	. . . 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 2598	. 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 4424	. 2 |- ( S Se A <-> A. x e. A { y e. A | y S x } e. _V )
9		df-se 4424	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
10	7, 8, 9	3imtr4g 205	1 |- ( R C_ S -> ( S Se A -> R Se A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   A.wral 2508   {crab 2512   _Vcvv 2799   C_ wss 3197   class class class wbr 4083   Se wse 4420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4202

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-br 4084  df-se 4424

This theorem is referenced by:  seeq1  4430