Theorem axltirr 8053

Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7952 with ordering on the extended reals. New proofs should use ltnr 8063 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)

Assertion

Ref	Expression
axltirr	⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)

Proof of Theorem axltirr

Step	Hyp	Ref	Expression
1		ax-pre-ltirr 7952	. 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
2		ltxrlt 8052	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐴 ↔ 𝐴 <ℝ 𝐴))
3	2	anidms 397	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐴 ↔ 𝐴 <ℝ 𝐴))
4	1, 3	mtbird 674	1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∈ wcel 2160   class class class wbr 4018  ℝcr 7839   <_ℝ cltrr 7844   < clt 8021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-pre-ltirr 7952

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-pnf 8023  df-mnf 8024  df-ltxr 8026

This theorem is referenced by:  ltnr  8063