Theorem dfpss4 3889

Description: Alternate definition of proper subset. Theorem IX.4.21 of [Rosser] p. 236. (Contributed by SF, 19-Jan-2015.)

Assertion

Ref	Expression
dfpss4	⊢ (A ⊊ B ↔ (A ⊆ B ∧ ∃x ∈ B ¬ x ∈ A))

Distinct variable groups:   x,A   x,B

Proof of Theorem dfpss4

Step	Hyp	Ref	Expression
1		dfpss3 3356	. 2 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ B ⊆ A))
2		dfss3 3264	. . . . 5 ⊢ (B ⊆ A ↔ ∀x ∈ B x ∈ A)
3		dfral2 2627	. . . . 5 ⊢ (∀x ∈ B x ∈ A ↔ ¬ ∃x ∈ B ¬ x ∈ A)
4	2, 3	bitr2i 241	. . . 4 ⊢ (¬ ∃x ∈ B ¬ x ∈ A ↔ B ⊆ A)
5	4	con1bii 321	. . 3 ⊢ (¬ B ⊆ A ↔ ∃x ∈ B ¬ x ∈ A)
6	5	anbi2i 675	. 2 ⊢ ((A ⊆ B ∧ ¬ B ⊆ A) ↔ (A ⊆ B ∧ ∃x ∈ B ¬ x ∈ A))
7	1, 6	bitri 240	1 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ∃x ∈ B ¬ x ∈ A))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   ⊆ wss 3258   ⊊ wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by:  ssfin  4471