Theorem elre0re 43307

Description: Specialized version of 0red 11312 without using ax-1cn 11259 and ax-cnre 11274. (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 11272	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11284	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2849	. . . 4 ⊢ ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
4	2, 3	syl5ibcom 248	. . 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 16	1 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  (class class class)co 7421  ℝcr 11200  0cc0 11201   + caddc 11204

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-addrcl 11262  ax-rnegex 11272

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-clel 2836  df-rex 3088

This theorem is used by:  redvmptabs  43413  rernegcl  43424  renegadd  43425  reneg0addlid  43427  resubeulem1  43428  resubeulem2  43429  resubeu  43430  remul02  43458  remul01  43460  readdrid  43463  resubid1  43464  renegneg  43465  renegid2  43467  sn-it0e0  43469  relt0neg2  43523