Theorem nsstr 39273

Description: If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
nsstr	|- ( ( -. A C_ B /\ C C_ B ) -> -. A C_ C )

Proof of Theorem nsstr

Step	Hyp	Ref	Expression
1		sstr 3611	. . . 4 |- ( ( A C_ C /\ C C_ B ) -> A C_ B )
2	1	ancoms 469	. . 3 |- ( ( C C_ B /\ A C_ C ) -> A C_ B )
3	2	adantll 750	. 2 |- ( ( ( -. A C_ B /\ C C_ B ) /\ A C_ C ) -> A C_ B )
4		simpll 790	. 2 |- ( ( ( -. A C_ B /\ C C_ B ) /\ A C_ C ) -> -. A C_ B )
5	3, 4	pm2.65da 600	1 |- ( ( -. A C_ B /\ C C_ B ) -> -. A C_ C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   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-in 3581  df-ss 3588

This theorem is referenced by:  mbfpsssmf  40991