Theorem elre0re 42875

Description: Specialized version of 0red 11195 without using ax-1cn 11142 and ax-cnre 11157. (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.

Step	Hyp	Ref	Expression
1		ax-rnegex 11155	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11167	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2851	. . . 4 ⊢ ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
4	2, 3	syl5ibcom 247	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
5	4	rexlimdva 3164	. 2 ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
6	1, 5	mpd 15	1 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∃wrex 3087  (class class class)co 7396  ℝcr 11083  0cc0 11084   + caddc 11087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-addrcl 11145  ax-rnegex 11155

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-cleq 2755  df-clel 2838  df-rex 3088

This theorem is referenced by:  redvmptabs  42974  rernegcl  42985  renegadd  42986  reneg0addlid  42988  resubeulem1  42989  resubeulem2  42990  resubeu  42991  remul02  43019  remul01  43021  readdrid  43024  resubid1  43025  renegneg  43026  renegid2  43028  sn-it0e0  43030  relt0neg2  43084