Theorem axltirr 8221

Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 8119 with ordering on the extended reals. New proofs should use ltnr 8231 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 8119	. 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
2		ltxrlt 8220	. . 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 677	1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∈ wcel 2200   class class class wbr 4083  ℝcr 8006   <_ℝ cltrr 8011   < clt 8189

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-pre-ltirr 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8191  df-mnf 8192  df-ltxr 8194

This theorem is referenced by:  ltnr  8231