Theorem elre0re 42681

Description: Specialized version of 0red 11136 without using ax-1cn 11085 and ax-cnre 11100. (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 11098	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11110	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2823	. . . 4 ⊢ ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
4	2, 3	syl5ibcom 245	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
5	4	rexlimdva 3136	. 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 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3059  (class class class)co 7356  ℝcr 11026  0cc0 11027   + caddc 11030

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-addrcl 11088  ax-rnegex 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2727  df-clel 2810  df-rex 3060

This theorem is referenced by:  redvmptabs  42780  rernegcl  42791  renegadd  42792  reneg0addlid  42794  resubeulem1  42795  resubeulem2  42796  resubeu  42797  remul02  42825  remul01  42827  readdrid  42830  resubid1  42831  renegneg  42832  renegid2  42834  sn-it0e0  42836  relt0neg2  42890