Theorem pwntru 4228

Description: A slight strengthening of pwtrufal 15488. (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 110	. . . 4 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A =/= { (/) } )
2	1	neneqd 2385	. . 3 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> -. A = { (/) } )
3		simpll 527	. . . . . 6 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> A C_ { (/) } )
4		simpl 109	. . . . . . . . . 10 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A C_ { (/) } )
5	4	sselda 3179	. . . . . . . . 9 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x e. { (/) } )
6		elsni 3636	. . . . . . . . 9 |- ( x e. { (/) } -> x = (/) )
7	5, 6	syl 14	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x = (/) )
8		simpr 110	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x e. A )
9	7, 8	eqeltrrd 2271	. . . . . . 7 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> (/) e. A )
10	9	snssd 3763	. . . . . 6 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> { (/) } C_ A )
11	3, 10	eqssd 3196	. . . . 5 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> A = { (/) } )
12	11	ex 115	. . . 4 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( x e. A -> A = { (/) } ) )
13	12	exlimdv 1830	. . 3 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( E. x x e. A -> A = { (/) } ) )
14	2, 13	mtod 664	. 2 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> -. E. x x e. A )
15		notm0 3467	. 2 |- ( -. E. x x e. A <-> A = (/) )
16	14, 15	sylib 122	1 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   =/= wne 2364   C_ wss 3153   (/)c0 3446   {csn 3618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624

This theorem is referenced by:  exmid1dc  4229  exmid1stab  4237