Theorem nelne1 3036

Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 14-May-2023.)

Assertion

Ref	Expression
nelne1	⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)

Proof of Theorem nelne1

Step	Hyp	Ref	Expression
1		nelneq2 2863	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)
2	1	neqned 2944	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2105   ≠ wne 2937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-clel 2813  df-ne 2938

This theorem is referenced by:  elnelne1  3054  difsnb  4810  fofinf1o  9369  fin23lem24  10359  fin23lem31  10380  ttukeylem7  10552  npomex  11033  lbspss  21098  islbs3  21174  lbsextlem4  21180  obslbs  21767  hauspwpwf1  24010  ppiltx  27234  tglineneq  28666  lnopp2hpgb  28785  colopp  28791  ex-pss  30456  drngidlhash  33441  ssdifidlprm  33465  mxidlmaxv  33475  mxidlprm  33477  drnglidl1ne0  33482  drng0mxidl  33483  qsdrnglem2  33503  rsprprmprmidl  33529  1arithufdlem4  33554  ply1annnr  33710  irngnminplynz  33719  algextdeglem4  33725  unelldsys  34138  cntnevol  34208  fin2solem  37592  lshpnelb  38965  osumcllem10N  39947  pexmidlem7N  39958  dochsnkrlem1  41451  ricdrng1  42514  rpnnen3lem  43019  lvecpsslmod  48352