Theorem axltirr 8393

Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 8292 with ordering on the extended reals. New proofs should use ltnr 8403 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 8292	. 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
2		ltxrlt 8392	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐴 ↔ 𝐴 <ℝ 𝐴))
3	2	anidms 401	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐴 ↔ 𝐴 <ℝ 𝐴))
4	1, 3	mtbird 684	1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105   ∈ wcel 2209   class class class wbr 4130  ℝcr 8179   <_ℝ cltrr 8184   < clt 8361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-pre-ltirr 8292

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-pnf 8363  df-mnf 8364  df-ltxr 8366

This theorem is used by:  ltnr  8403