Theorem negm 9738

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 3187	. . . . 5 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
2		renegcl 8335	. . . . . . . 8 |- ( x e. RR -> -u x e. RR )
3		negeq 8267	. . . . . . . . . 10 |- ( z = -u x -> -u z = -u -u x )
4	3	eleq1d 2274	. . . . . . . . 9 |- ( z = -u x -> ( -u z e. A <-> -u -u x e. A ) )
5	4	elrab3 2930	. . . . . . . 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 8060	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8	7	negnegd 8376	. . . . . . . 8 |- ( x e. RR -> -u -u x = x )
9	8	eleq1d 2274	. . . . . . 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 2788	. . . 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 1842	. 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 1373   E.wex 1515   e. wcel 2176   {crab 2488   C_ wss 3166   RRcr 7926   -ucneg 8246

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586  ax-resscn 8019  ax-1cn 8020  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-sub 8247  df-neg 8248

This theorem is referenced by: (None)