Theorem elre0re 43225

Description: Specialized version of 0red 11282 without using ax-1cn 11229 and ax-cnre 11244. (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 11242	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11254	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2848	. . . 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 3163	. 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 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  (class class class)co 7408  ℝcr 11170  0cc0 11171   + caddc 11174

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-addrcl 11232  ax-rnegex 11242

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-clel 2835  df-rex 3087

This theorem is used by:  redvmptabs  43339  rernegcl  43350  renegadd  43351  reneg0addlid  43353  resubeulem1  43354  resubeulem2  43355  resubeu  43356  remul02  43384  remul01  43386  readdrid  43389  resubid1  43390  renegneg  43391  renegid2  43393  sn-it0e0  43395  relt0neg2  43449