Theorem neldif 1594

Description: Implication of membership in a class difference.

Assertion

Ref	Expression
neldif	⊢ ((A ∈ B ∧ ¬ A ∈ (B ∖ C)) → A ∈ C)

Proof of Theorem neldif

Step	Hyp	Ref	Expression
1		eldif 1496	. . . . 5 ⊢ (A ∈ (B ∖ C) ↔ (A ∈ B ∧ ¬ A ∈ C))
2	1	biimpr 134	. . . 4 ⊢ ((A ∈ B ∧ ¬ A ∈ C) → A ∈ (B ∖ C))
3	2	exp 291	. . 3 ⊢ (A ∈ B → (¬ A ∈ C → A ∈ (B ∖ C)))
4	3	con1d 85	. 2 ⊢ (A ∈ B → (¬ A ∈ (B ∖ C) → A ∈ C))
5	4	imp 277	1 ⊢ ((A ∈ B ∧ ¬ A ∈ (B ∖ C)) → A ∈ C)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wcel 1092   ∖ cdif 1484

This theorem is referenced by:  peano5 2394

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489

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