Theorem sspssn 3048

Description: Like pssn2lp 3045 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)

Assertion

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

Proof of Theorem sspssn

Step	Hyp	Ref	Expression
1		pm3.24 627	. 2 |- -. ( B C. A /\ -. B C. A )
2		ssnpss 3047	. . . 4 |- ( A C_ B -> -. B C. A )
3	2	anim2i 324	. . 3 |- ( ( B C. A /\ A C_ B ) -> ( B C. A /\ -. B C. A ) )
4	3	ancoms 255	. 2 |- ( ( A C_ B /\ B C. A ) -> ( B C. A /\ -. B C. A ) )
5	1, 4	mto 588	1 |- -. ( A C_ B /\ B C. A )

Syntax hints:   -. wn 3   /\ wa 97   C_ wss 2917   C. wpss 2918

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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by:  sspsstr  3050  psssstr  3051