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Theorem nelrnres 38883
Description: If 𝐴 is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
nelrnres 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵𝐶))

Proof of Theorem nelrnres
StepHypRef Expression
1 rnresss 38874 . 2 ran (𝐵𝐶) ⊆ ran 𝐵
2 ssnel 38726 . 2 ((ran (𝐵𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵𝐶))
31, 2mpan 705 1 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1987  wss 3560  ran crn 5085  cres 5086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096
This theorem is referenced by:  sge0sup  39945  sge0less  39946  sge0resplit  39960
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