Theorem npss 4038

Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3930. (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		pm4.61 408	. . 3 ⊢ (¬ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
2		dfpss2 4013	. . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
3	1, 2	bitr4i 281	. 2 ⊢ (¬ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵) ↔ 𝐴 ⊊ 𝐵)
4	3	con1bii 360	1 ⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ⊆ wss 3881   ⊊ wpss 3882

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

This theorem depends on definitions:  df-bi 210  df-an 400  df-ne 2988  df-pss 3900

This theorem is referenced by:  ttukeylem7  9926  canthp1lem2  10064  pgpfac1lem1  19189  lspsncv0  19911  obslbs  20419  ssmxidl  31050  fvineqsneq  34829