Theorem npss 3379

Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3287. (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 3354	. . 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3257   ⊊ wpss 3258

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-ne 2518  df-pss 3261

This theorem is referenced by: (None)