Theorem pwntru 4312

Description: A slight strengthening of pwtrufal 16771. (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 2433	. . 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 3238	. . . . . . . . 9 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> x e. { (/) } )
6		elsni 3707	. . . . . . . . 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 2310	. . . . . . 7 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> (/) e. A )
10	9	snssd 3839	. . . . . 6 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> { (/) } C_ A )
11	3, 10	eqssd 3255	. . . . 5 |- ( ( ( A C_ { (/) } /\ A =/= { (/) } ) /\ x e. A ) -> A = { (/) } )
12	11	ex 115	. . . 4 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( x e. A -> A = { (/) } ) )
13	12	exlimdv 1868	. . 3 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> ( E. x x e. A -> A = { (/) } ) )
14	2, 13	mtod 669	. 2 |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> -. E. x x e. A )
15		notm0 3529	. 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 1398   E.wex 1541   e. wcel 2203   =/= wne 2412   C_ wss 3211   (/)c0 3508   {csn 3689

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695

This theorem is referenced by:  exmid1dc  4313  exmid1stab  4321  pw1ndom3lem  16763