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Theorem axltirr 7497
Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7401 with ordering on the extended reals. New proofs should use ltnr 7506 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 7401 . 2 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2 ltxrlt 7496 . . 3 ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐴𝐴 < 𝐴))
32anidms 389 . 2 (𝐴 ∈ ℝ → (𝐴 < 𝐴𝐴 < 𝐴))
41, 3mtbird 631 1 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wcel 1436   class class class wbr 3820  cr 7293   < cltrr 7298   < clt 7466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-pre-ltirr 7401
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-xp 4417  df-pnf 7468  df-mnf 7469  df-ltxr 7471
This theorem is referenced by:  ltnr  7506
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