Theorem npss 3380

Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3288. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
npss	⊢ (¬ A ⊊ B ↔ (A ⊆ B → A = B))

Proof of Theorem npss

Step	Hyp	Ref	Expression
1		pm4.61 415	. . 3 ⊢ (¬ (A ⊆ B → A = B) ↔ (A ⊆ B ∧ ¬ A = B))
2		dfpss2 3355	. . 3 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ A = B))
3	1, 2	bitr4i 243	. 2 ⊢ (¬ (A ⊆ B → A = B) ↔ A ⊊ B)
4	3	con1bii 321	1 ⊢ (¬ A ⊊ B ↔ (A ⊆ B → A = B))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3258   ⊊ wpss 3259

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-ne 2519  df-pss 3262

This theorem is used by: (None)