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Theorem elre0re 39998
Description: Specialized version of 0red 10836 without using ax-1cn 10787 and ax-cnre 10802. (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 10800 . 2 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2 readdcl 10812 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3 eleq1 2825 . . . 4 ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
42, 3syl5ibcom 248 . . 3 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
54rexlimdva 3203 . 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 1543  wcel 2110  wrex 3062  (class class class)co 7213  cr 10728  0cc0 10729   + caddc 10732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-addrcl 10790  ax-rnegex 10800
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067
This theorem is referenced by:  rernegcl  40062  renegadd  40063  reneg0addid2  40065  resubeulem1  40066  resubeulem2  40067  resubeu  40068  remul02  40096  remul01  40098  readdid1  40100  resubid1  40101  renegneg  40102  renegid2  40104  sn-it0e0  40105  relt0neg2  40134  sn-inelr  40143
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