Theorem negm 9994

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 3242	. . . . 5 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
2		renegcl 8577	. . . . . . . 8 |- ( x e. RR -> -u x e. RR )
3		negeq 8509	. . . . . . . . . 10 |- ( z = -u x -> -u z = -u -u x )
4	3	eleq1d 2307	. . . . . . . . 9 |- ( z = -u x -> ( -u z e. A <-> -u -u x e. A ) )
5	4	elrab3 2983	. . . . . . . 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 8302	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8	7	negnegd 8618	. . . . . . . 8 |- ( x e. RR -> -u -u x = x )
9	8	eleq1d 2307	. . . . . . 7 |- ( x e. RR -> ( -u -u x e. A <-> x e. A ) )
10	6, 9	bitrd 188	. . . . . 6 |- ( x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> x e. A ) )
11	10	biimprd 158	. . . . 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 2838	. . . 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 1872	. 2 |- ( A C_ RR -> ( E. x x e. A -> E. y y e. { z e. RR | -u z e. A } ) )
16	15	imp 124	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 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   C_ wss 3220   RRcr 8168   -ucneg 8488

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-neg 8490

This theorem is referenced by: (None)