Theorem nssne1 3993

Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Assertion

Ref	Expression
nssne1	⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 ⊆ 𝐶) → 𝐵 ≠ 𝐶)

Proof of Theorem nssne1

Step	Hyp	Ref	Expression
1		sseq2 3957	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶))
2	1	biimpcd 252	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 = 𝐶 → 𝐴 ⊆ 𝐶))
3	2	necon3bd 2970	. 2 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊆ 𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 412	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 ⊆ 𝐶) → 𝐵 ≠ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2956   ⊆ wss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-ne 2957  df-ss 3916

This theorem is used by: (None)