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Theorem minel 3260
Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
Assertion
Ref Expression
minel ((A B (𝐶B) = ∅) → ¬ A 𝐶)

Proof of Theorem minel
StepHypRef Expression
1 inelcm 3259 . . . . 5 ((A 𝐶 A B) → (𝐶B) ≠ ∅)
21necon2bi 2238 . . . 4 ((𝐶B) = ∅ → ¬ (A 𝐶 A B))
3 imnan 611 . . . 4 ((A 𝐶 → ¬ A B) ↔ ¬ (A 𝐶 A B))
42, 3sylibr 137 . . 3 ((𝐶B) = ∅ → (A 𝐶 → ¬ A B))
54con2d 542 . 2 ((𝐶B) = ∅ → (A B → ¬ A 𝐶))
65impcom 116 1 ((A B (𝐶B) = ∅) → ¬ A 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1228   wcel 1374  cin 2893  c0 3201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-in 2901  df-nul 3202
This theorem is referenced by: (None)
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