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Theorem axltirr 7749
Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7651 with ordering on the extended reals. New proofs should use ltnr 7758 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 7651 . 2 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2 ltxrlt 7748 . . 3 ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐴𝐴 < 𝐴))
32anidms 392 . 2 (𝐴 ∈ ℝ → (𝐴 < 𝐴𝐴 < 𝐴))
41, 3mtbird 645 1 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wcel 1461   class class class wbr 3893  cr 7540   < cltrr 7545   < clt 7718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-pre-ltirr 7651
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-xp 4503  df-pnf 7720  df-mnf 7721  df-ltxr 7723
This theorem is referenced by:  ltnr  7758
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