Theorem elre0re 43050

Description: Specialized version of 0red 11217 without using ax-1cn 11164 and ax-cnre 11179. (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 11177	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11189	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2850	. . . 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 3165	. 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 400   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  (class class class)co 7412  ℝcr 11105  0cc0 11106   + caddc 11109

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-addrcl 11167  ax-rnegex 11177

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-clel 2837  df-rex 3089

This theorem is used by:  redvmptabs  43149  rernegcl  43160  renegadd  43161  reneg0addlid  43163  resubeulem1  43164  resubeulem2  43165  resubeu  43166  remul02  43194  remul01  43196  readdrid  43199  resubid1  43200  renegneg  43201  renegid2  43203  sn-it0e0  43205  relt0neg2  43259