Theorem pwtrufal 15206

Description: A subset of the singleton { (/) } cannot be anything other than (/) or { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4216. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4214), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)

Assertion

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

Proof of Theorem pwtrufal

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 531	. . . . 5 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> -. A = { (/) } )
2		simpll 527	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> A C_ { (/) } )
3		simpl 109	. . . . . . . . . . . 12 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> A C_ { (/) } )
4	3	sselda 3170	. . . . . . . . . . 11 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> x e. { (/) } )
5		elsni 3625	. . . . . . . . . . 11 |- ( x e. { (/) } -> x = (/) )
6	4, 5	syl 14	. . . . . . . . . 10 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> x = (/) )
7		simpr 110	. . . . . . . . . 10 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> x e. A )
8	6, 7	eqeltrrd 2267	. . . . . . . . 9 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> (/) e. A )
9	8	snssd 3752	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> { (/) } C_ A )
10	2, 9	eqssd 3187	. . . . . . 7 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> A = { (/) } )
11	10	ex 115	. . . . . 6 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> ( x e. A -> A = { (/) } ) )
12	11	exlimdv 1830	. . . . 5 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> ( E. x x e. A -> A = { (/) } ) )
13	1, 12	mtod 664	. . . 4 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> -. E. x x e. A )
14		notm0 3458	. . . 4 |- ( -. E. x x e. A <-> A = (/) )
15	13, 14	sylib 122	. . 3 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> A = (/) )
16		simprl 529	. . 3 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> -. A = (/) )
17	15, 16	pm2.65da 662	. 2 |- ( A C_ { (/) } -> -. ( -. A = (/) /\ -. A = { (/) } ) )
18		ioran 753	. 2 |- ( -. ( A = (/) \/ A = { (/) } ) <-> ( -. A = (/) /\ -. A = { (/) } ) )
19	17, 18	sylnibr 678	1 |- ( A C_ { (/) } -> -. -. ( A = (/) \/ A = { (/) } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2160   C_ wss 3144   (/)c0 3437   {csn 3607

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613

This theorem is referenced by:  pwle2  15207