Theorem elre0re 43348

Description: Specialized version of 0red 11311 without using ax-1cn 11258 and ax-cnre 11273. (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 11271	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 11283	. . . 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 1569   ∈ wcel 2144  ∃wrex 3086  (class class class)co 7420  ℝcr 11199  0cc0 11200   + caddc 11203

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-addrcl 11261  ax-rnegex 11271

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-clel 2835  df-rex 3087

This theorem is used by:  redvmptabs  43454  rernegcl  43465  renegadd  43466  reneg0addlid  43468  resubeulem1  43469  resubeulem2  43470  resubeu  43471  remul02  43499  remul01  43501  readdrid  43504  resubid1  43505  renegneg  43506  renegid2  43508  sn-it0e0  43510  relt0neg2  43564