Theorem psstr 3374

Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
psstr	⊢ ((A ⊊ B ∧ B ⊊ C) → A ⊊ C)

Proof of Theorem psstr

Step	Hyp	Ref	Expression
1		pssss 3365	. . 3 ⊢ (A ⊊ B → A ⊆ B)
2		pssss 3365	. . 3 ⊢ (B ⊊ C → B ⊆ C)
3	1, 2	sylan9ss 3286	. 2 ⊢ ((A ⊊ B ∧ B ⊊ C) → A ⊆ C)
4		pssn2lp 3371	. . . 4 ⊢ ¬ (C ⊊ B ∧ B ⊊ C)
5		psseq1 3357	. . . . 5 ⊢ (A = C → (A ⊊ B ↔ C ⊊ B))
6	5	anbi1d 685	. . . 4 ⊢ (A = C → ((A ⊊ B ∧ B ⊊ C) ↔ (C ⊊ B ∧ B ⊊ C)))
7	4, 6	mtbiri 294	. . 3 ⊢ (A = C → ¬ (A ⊊ B ∧ B ⊊ C))
8	7	con2i 112	. 2 ⊢ ((A ⊊ B ∧ B ⊊ C) → ¬ A = C)
9		dfpss2 3355	. 2 ⊢ (A ⊊ C ↔ (A ⊆ C ∧ ¬ A = C))
10	3, 8, 9	sylanbrc 645	1 ⊢ ((A ⊊ B ∧ B ⊊ C) → A ⊊ C)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ⊆ 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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by:  sspsstr  3375  psssstr  3376  psstrd  3377