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Theorem minel 3277
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 3276 . . . . 5 ((A 𝐶 A B) → (𝐶B) ≠ ∅)
21necon2bi 2254 . . . 4 ((𝐶B) = ∅ → ¬ (A 𝐶 A B))
3 imnan 623 . . . 4 ((A 𝐶 → ¬ A B) ↔ ¬ (A 𝐶 A B))
42, 3sylibr 137 . . 3 ((𝐶B) = ∅ → (A 𝐶 → ¬ A B))
54con2d 554 . 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 1242   wcel 1390  cin 2910  c0 3218
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219
This theorem is referenced by: (None)
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