Theorem negm 9553

Description: The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)

Assertion

Ref	Expression
negm	|- ( ( A C_ RR /\ E. x x e. A ) -> E. y y e. { z e. RR | -u z e. A } )

Distinct variable group:   x, A, y, z

Proof of Theorem negm

Step	Hyp	Ref	Expression
1		ssel 3136	. . . . 5 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
2		renegcl 8159	. . . . . . . 8 |- ( x e. RR -> -u x e. RR )
3		negeq 8091	. . . . . . . . . 10 |- ( z = -u x -> -u z = -u -u x )
4	3	eleq1d 2235	. . . . . . . . 9 |- ( z = -u x -> ( -u z e. A <-> -u -u x e. A ) )
5	4	elrab3 2883	. . . . . . . 8 |- ( -u x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> -u -u x e. A ) )
6	2, 5	syl 14	. . . . . . 7 |- ( x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> -u -u x e. A ) )
7		recn 7886	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8	7	negnegd 8200	. . . . . . . 8 |- ( x e. RR -> -u -u x = x )
9	8	eleq1d 2235	. . . . . . 7 |- ( x e. RR -> ( -u -u x e. A <-> x e. A ) )
10	6, 9	bitrd 187	. . . . . 6 |- ( x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> x e. A ) )
11	10	biimprd 157	. . . . 5 |- ( x e. RR -> ( x e. A -> -u x e. { z e. RR | -u z e. A } ) )
12	1, 11	syli 37	. . . 4 |- ( A C_ RR -> ( x e. A -> -u x e. { z e. RR | -u z e. A } ) )
13		elex2 2742	. . . 4 |- ( -u x e. { z e. RR | -u z e. A } -> E. y y e. { z e. RR | -u z e. A } )
14	12, 13	syl6 33	. . 3 |- ( A C_ RR -> ( x e. A -> E. y y e. { z e. RR | -u z e. A } ) )
15	14	exlimdv 1807	. 2 |- ( A C_ RR -> ( E. x x e. A -> E. y y e. { z e. RR | -u z e. A } ) )
16	15	imp 123	1 |- ( ( A C_ RR /\ E. x x e. A ) -> E. y y e. { z e. RR | -u z e. A } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   {crab 2448   C_ wss 3116   RRcr 7752   -ucneg 8070

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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-neg 8072

This theorem is referenced by: (None)