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Theorem axltirr 7086
 Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6996 with ordering on the extended reals. New proofs should use ltnr 7095 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
StepHypRef Expression
1 ax-pre-ltirr 6996 . 2 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2 ltxrlt 7085 . . 3 ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐴𝐴 < 𝐴))
32anidms 377 . 2 (𝐴 ∈ ℝ → (𝐴 < 𝐴𝐴 < 𝐴))
41, 3mtbird 598 1 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∈ wcel 1393   class class class wbr 3764  ℝcr 6888   <ℝ cltrr 6893   < clt 7060 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-ltxr 7065 This theorem is referenced by:  ltnr  7095
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