Theorem elre0re 39998

Description: Specialized version of 0red 10836 without using ax-1cn 10787 and ax-cnre 10802. (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 10800	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 10812	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2825	. . . 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 3203	. 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 1543   ∈ wcel 2110  ∃wrex 3062  (class class class)co 7213  ℝcr 10728  0cc0 10729   + caddc 10732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-addrcl 10790  ax-rnegex 10800

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067

This theorem is referenced by:  rernegcl  40062  renegadd  40063  reneg0addid2  40065  resubeulem1  40066  resubeulem2  40067  resubeu  40068  remul02  40096  remul01  40098  readdid1  40100  resubid1  40101  renegneg  40102  renegid2  40104  sn-it0e0  40105  relt0neg2  40134  sn-inelr  40143