Theorem elre0re 42968

Description: Specialized version of 0red 11210 without using ax-1cn 11157 and ax-cnre 11172. (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 11170	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11182	. . . 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  (class class class)co 7410  ℝcr 11098  0cc0 11099   + caddc 11102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-addrcl 11160  ax-rnegex 11170

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-cleq 2753  df-clel 2836  df-rex 3088

This theorem is referenced by:  redvmptabs  43067  rernegcl  43078  renegadd  43079  reneg0addlid  43081  resubeulem1  43082  resubeulem2  43083  resubeu  43084  remul02  43112  remul01  43114  readdrid  43117  resubid1  43118  renegneg  43119  renegid2  43121  sn-it0e0  43123  relt0neg2  43177