Theorem pwntru 4160

Description: A slight strengthening of pwtrufal 13540. (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 2348	. . 3 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> -. A = { (/) } )
3		simpll 519	. . . . . 6 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> A C_ { (/) } )
4		simpl 108	. . . . . . . . . 10 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A C_ { (/) } )
5	4	sselda 3128	. . . . . . . . 9 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x e. { (/) } )
6		elsni 3578	. . . . . . . . 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 2235	. . . . . . 7 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> (/) e. A )
10	9	snssd 3701	. . . . . 6 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> { (/) } C_ A )
11	3, 10	eqssd 3145	. . . . 5 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> A = { (/) } )
12	11	ex 114	. . . 4 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( x e. A -> A = { (/) } ) )
13	12	exlimdv 1799	. . 3 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( E. x x e. A -> A = { (/) } ) )
14	2, 13	mtod 653	. 2 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> -. E. x x e. A )
15		notm0 3414	. 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 1335   E.wex 1472   e. wcel 2128   =/= wne 2327   C_ wss 3102   (/)c0 3394   {csn 3560

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566

This theorem is referenced by:  exmid1dc  4161  exmid1stab  13543