Theorem elre0re 42808

Description: Specialized version of 0red 11170 without using ax-1cn 11117 and ax-cnre 11132. (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 11130	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11142	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2840	. . . 4 ⊢ ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
4	2, 3	syl5ibcom 247	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
5	4	rexlimdva 3153	. 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 398   = wceq 1550   ∈ wcel 2132  ∃wrex 3076  (class class class)co 7381  ℝcr 11058  0cc0 11059   + caddc 11062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-addrcl 11120  ax-rnegex 11130

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-cleq 2744  df-clel 2827  df-rex 3077

This theorem is referenced by:  redvmptabs  42907  rernegcl  42918  renegadd  42919  reneg0addlid  42921  resubeulem1  42922  resubeulem2  42923  resubeu  42924  remul02  42952  remul01  42954  readdrid  42957  resubid1  42958  renegneg  42959  renegid2  42961  sn-it0e0  42963  relt0neg2  43017