Theorem nssne1 3661

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 3627	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶))
2	1	biimpcd 239	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 = 𝐶 → 𝐴 ⊆ 𝐶))
3	2	necon3bd 2808	. 2 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊆ 𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 445	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 ⊆ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ≠ wne 2794   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-in 3581  df-ss 3588

This theorem is referenced by: (None)