Theorem nssne1 3055

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 3021	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶))
2	1	biimpcd 157	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 = 𝐶 → 𝐴 ⊆ 𝐶))
3	2	necon3bd 2288	. 2 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊆ 𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 122	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 ⊆ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   = wceq 1284   ≠ wne 2245   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ne 2246  df-in 2979  df-ss 2986

This theorem is referenced by: (None)