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Theorem elre0re 41477
Description: Specialized version of 0red 11221 without using ax-1cn 11170 and ax-cnre 11185. (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 11183 . 2 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2 readdcl 11195 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3 eleq1 2821 . . . 4 ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
42, 3syl5ibcom 244 . . 3 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
54rexlimdva 3155 . 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 396   = wceq 1541  wcel 2106  wrex 3070  (class class class)co 7411  cr 11111  0cc0 11112   + caddc 11115
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-addrcl 11173  ax-rnegex 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724  df-clel 2810  df-rex 3071
This theorem is referenced by:  rernegcl  41546  renegadd  41547  reneg0addlid  41549  resubeulem1  41550  resubeulem2  41551  resubeu  41552  remul02  41580  remul01  41582  readdrid  41584  resubid1  41585  renegneg  41586  renegid2  41588  sn-it0e0  41590  relt0neg2  41620
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