Theorem pwtrufal 13877

Description: A subset of the singleton { (/) } cannot be anything other than (/) or { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4177. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4175), 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 522	. . . . 5 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> -. A = { (/) } )
2		simpll 519	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> A C_ { (/) } )
3		simpl 108	. . . . . . . . . . . 12 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> A C_ { (/) } )
4	3	sselda 3142	. . . . . . . . . . 11 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> x e. { (/) } )
5		elsni 3594	. . . . . . . . . . 11 |- ( x e. { (/) } -> x = (/) )
6	4, 5	syl 14	. . . . . . . . . 10 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> x = (/) )
7		simpr 109	. . . . . . . . . 10 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> x e. A )
8	6, 7	eqeltrrd 2244	. . . . . . . . 9 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> (/) e. A )
9	8	snssd 3718	. . . . . . . 8 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> { (/) } C_ A )
10	2, 9	eqssd 3159	. . . . . . 7 |- ( ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) /\ x e. A ) -> A = { (/) } )
11	10	ex 114	. . . . . 6 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> ( x e. A -> A = { (/) } ) )
12	11	exlimdv 1807	. . . . 5 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> ( E. x x e. A -> A = { (/) } ) )
13	1, 12	mtod 653	. . . 4 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> -. E. x x e. A )
14		notm0 3429	. . . 4 |- ( -. E. x x e. A <-> A = (/) )
15	13, 14	sylib 121	. . 3 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> A = (/) )
16		simprl 521	. . 3 |- ( ( A C_ { (/) } /\ ( -. A = (/) /\ -. A = { (/) } ) ) -> -. A = (/) )
17	15, 16	pm2.65da 651	. 2 |- ( A C_ { (/) } -> -. ( -. A = (/) /\ -. A = { (/) } ) )
18		ioran 742	. 2 |- ( -. ( A = (/) \/ A = { (/) } ) <-> ( -. A = (/) /\ -. A = { (/) } ) )
19	17, 18	sylnibr 667	1 |- ( A C_ { (/) } -> -. -. ( A = (/) \/ A = { (/) } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1343   E.wex 1480   e. wcel 2136   C_ wss 3116   (/)c0 3409   {csn 3576

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582

This theorem is referenced by:  pwle2  13878