Theorem difsnpssim 3507

Description: ( B \ { A } ) is a proper subclass of B if A is a member of B. In classical logic, the converse holds as well. (Contributed by Jim Kingdon, 9-Aug-2018.)

Assertion

Ref	Expression
difsnpssim	|- ( A e. B -> ( B \ { A } ) C. B )

Proof of Theorem difsnpssim

Step	Hyp	Ref	Expression
1		notnot 559	. 2 |- ( A e. B -> -. -. A e. B )
2		difss 3070	. . . 4 |- ( B \ { A } ) C_ B
3	2	biantrur 287	. . 3 |- ( ( B \ { A } ) =/= B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
4		difsnb 3506	. . . 4 |- ( -. A e. B <-> ( B \ { A } ) = B )
5	4	necon3bbii 2242	. . 3 |- ( -. -. A e. B <-> ( B \ { A } ) =/= B )
6		df-pss 2933	. . 3 |- ( ( B \ { A } ) C. B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
7	3, 5, 6	3bitr4i 201	. 2 |- ( -. -. A e. B <-> ( B \ { A } ) C. B )
8	1, 7	sylib 127	1 |- ( A e. B -> ( B \ { A } ) C. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   e. wcel 1393   =/= wne 2204   \ cdif 2914   C_ wss 2917   C. wpss 2918   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-pss 2933  df-sn 3381

This theorem is referenced by: (None)