Theorem gruss 9618

Description: Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruss	|- ( ( U e. Univ /\ A e. U /\ B C_ A ) -> B e. U )

Proof of Theorem gruss

Step	Hyp	Ref	Expression
1		elpw2g 4827	. . . 4 |- ( A e. U -> ( B e. ~P A <-> B C_ A ) )
2	1	adantl 482	. . 3 |- ( ( U e. Univ /\ A e. U ) -> ( B e. ~P A <-> B C_ A ) )
3		grupw 9617	. . . . 5 |- ( ( U e. Univ /\ A e. U ) -> ~P A e. U )
4		gruelss 9616	. . . . 5 |- ( ( U e. Univ /\ ~P A e. U ) -> ~P A C_ U )
5	3, 4	syldan 487	. . . 4 |- ( ( U e. Univ /\ A e. U ) -> ~P A C_ U )
6	5	sseld 3602	. . 3 |- ( ( U e. Univ /\ A e. U ) -> ( B e. ~P A -> B e. U ) )
7	2, 6	sylbird 250	. 2 |- ( ( U e. Univ /\ A e. U ) -> ( B C_ A -> B e. U ) )
8	7	3impia 1261	1 |- ( ( U e. Univ /\ A e. U /\ B C_ A ) -> B e. U )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   Univcgru 9612

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-3an 1039  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-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-tr 4753  df-iota 5851  df-fv 5896  df-ov 6653  df-gru 9613

This theorem is referenced by:  grurn  9623  gruima  9624  gruxp  9629  grumap  9630  gruixp  9631  gruiin  9632  grudomon  9639  gruina  9640