Theorem nssrex 39260

Description: Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

Ref	Expression
nssrex	|- ( -. A C_ B <-> E. x e. A -. x e. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem nssrex

Step	Hyp	Ref	Expression
1		nss 3663	. 2 |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) )
2		df-rex 2918	. 2 |- ( E. x e. A -. x e. B <-> E. x ( x e. A /\ -. x e. B ) )
3	1, 2	bitr4i 267	1 |- ( -. A C_ B <-> E. x e. A -. x e. B )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   E.wrex 2913   C_ wss 3574

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

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-rex 2918  df-in 3581  df-ss 3588

This theorem is referenced by:  mapssbi  39405