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Theorem elre0re 39633
 Description: Specialized version of 0red 10651 without using ax-1cn 10602 and ax-cnre 10617. (Contributed by Steven Nguyen, 28-Jan-2023.)
Assertion
Ref Expression
elre0re (𝐴 ∈ ℝ → 0 ∈ ℝ)

Proof of Theorem elre0re
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ax-rnegex 10615 . 2 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2 readdcl 10627 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3 eleq1 2877 . . . 4 ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
42, 3syl5ibcom 248 . . 3 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
54rexlimdva 3244 . 2 (𝐴 ∈ ℝ → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
61, 5mpd 15 1 (𝐴 ∈ ℝ → 0 ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  (class class class)co 7145  ℝcr 10543  0cc0 10544   + caddc 10547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-addrcl 10605  ax-rnegex 10615 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112 This theorem is referenced by:  rernegcl  39680  renegadd  39681  reneg0addid2  39683  resubeulem1  39684  resubeulem2  39685  resubeu  39686  remul02  39714  remul01  39716  readdid1  39718  resubid1  39719  renegneg  39720  renegid2  39722  sn-it0e0  39723  relt0neg2  39752  sn-inelr  39761
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