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Theorem renegcli 7336
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7335 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypothesis
Ref Expression
renegcl.1 𝐴 ∈ ℝ
Assertion
Ref Expression
renegcli -𝐴 ∈ ℝ

Proof of Theorem renegcli
StepHypRef Expression
1 renegcl.1 . 2 𝐴 ∈ ℝ
2 renegcl 7335 . 2 (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
31, 2ax-mp 7 1 -𝐴 ∈ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1409  cr 6946  -cneg 7246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-neg 7248
This theorem is referenced by:  resubcli  7337  inelr  7649  cju  7989  neg1rr  8096  ex-fl  10279
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