Theorem nssne2 3982

Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Assertion

Ref	Expression
nssne2	⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵)

Proof of Theorem nssne2

Step	Hyp	Ref	Expression
1		sseq1 3946	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
2	1	biimpcd 248	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 = 𝐵 → 𝐵 ⊆ 𝐶))
3	2	necon3bd 2957	. 2 ⊢ (𝐴 ⊆ 𝐶 → (¬ 𝐵 ⊆ 𝐶 → 𝐴 ≠ 𝐵))
4	3	imp 407	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ≠ wne 2943   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  atcvatlem  30747  mdsymlem3  30767  disjdifprg  30914  mapdh6aN  39749  mapdh8e  39798  hdmap1l6a  39823