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Theorem nltled 8077
Description: 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
ltd.1 (𝜑𝐴 ∈ ℝ)
ltd.2 (𝜑𝐵 ∈ ℝ)
nltled.1 (𝜑 → ¬ 𝐵 < 𝐴)
Assertion
Ref Expression
nltled (𝜑𝐴𝐵)

Proof of Theorem nltled
StepHypRef Expression
1 nltled.1 . 2 (𝜑 → ¬ 𝐵 < 𝐴)
2 ltd.1 . . 3 (𝜑𝐴 ∈ ℝ)
3 ltd.2 . . 3 (𝜑𝐵 ∈ ℝ)
42, 3lenltd 8074 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
51, 4mpbird 167 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 2148   class class class wbr 4003  cr 7809   < clt 7991  cle 7992
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-xr 7995  df-le 7997
This theorem is referenced by:  ltntri  8084  suprubex  8907  infregelbex  9597  cvgratz  11539  zsupcl  11947  zssinfcl  11948  infssuzledc  11950  dvdslegcd  11964  pw2dvdseulemle  12166  dedekindeulemuub  14065  dedekindeulemlu  14069  suplociccex  14073  dedekindicclemuub  14074  dedekindicclemlu  14078  ivthinclemlopn  14084  ivthinclemuopn  14086  refeq  14746
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