Theorem pwntru 4185

Description: A slight strengthening of pwtrufal 14030. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)

Assertion

Ref	Expression
pwntru	|- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )

Proof of Theorem pwntru

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A =/= { (/) } )
2	1	neneqd 2361	. . 3 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> -. A = { (/) } )
3		simpll 524	. . . . . 6 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> A C_ { (/) } )
4		simpl 108	. . . . . . . . . 10 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A C_ { (/) } )
5	4	sselda 3147	. . . . . . . . 9 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x e. { (/) } )
6		elsni 3601	. . . . . . . . 9 |- ( x e. { (/) } -> x = (/) )
7	5, 6	syl 14	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x = (/) )
8		simpr 109	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x e. A )
9	7, 8	eqeltrrd 2248	. . . . . . 7 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> (/) e. A )
10	9	snssd 3725	. . . . . 6 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> { (/) } C_ A )
11	3, 10	eqssd 3164	. . . . 5 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> A = { (/) } )
12	11	ex 114	. . . 4 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( x e. A -> A = { (/) } ) )
13	12	exlimdv 1812	. . 3 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( E. x x e. A -> A = { (/) } ) )
14	2, 13	mtod 658	. 2 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> -. E. x x e. A )
15		notm0 3435	. 2 |- ( -. E. x x e. A <-> A = (/) )
16	14, 15	sylib 121	1 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   =/= wne 2340   C_ wss 3121   (/)c0 3414   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589

This theorem is referenced by:  exmid1dc  4186  exmid1stab  14033