Theorem nssne2 3991

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 3955	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
2	1	biimpcd 248	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 = 𝐵 → 𝐵 ⊆ 𝐶))
3	2	necon3bd 2955	. 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 1540   ≠ wne 2941   ⊆ wss 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-v 3443  df-in 3903  df-ss 3913

This theorem is referenced by:  atcvatlem  30855  mdsymlem3  30875  disjdifprg  31022  mapdh6aN  39961  mapdh8e  40010  hdmap1l6a  40035