Theorem nssne2 3976

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 3940	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
2	1	biimpcd 252	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 = 𝐵 → 𝐵 ⊆ 𝐶))
3	2	necon3bd 3001	. 2 ⊢ (𝐴 ⊆ 𝐶 → (¬ 𝐵 ⊆ 𝐶 → 𝐴 ≠ 𝐵))
4	3	imp 410	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ≠ wne 2987   ⊆ wss 3881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-in 3888  df-ss 3898

This theorem is referenced by:  atcvatlem  30168  mdsymlem3  30188  disjdifprg  30338  mapdh6aN  39031  mapdh8e  39080  hdmap1l6a  39105