Theorem psstr 4061

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 4051	. . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
2		pssss 4051	. . 3 ⊢ (𝐵 ⊊ 𝐶 → 𝐵 ⊆ 𝐶)
3	1, 2	sylan9ss 3949	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊆ 𝐶)
4		pssn2lp 4058	. . . 4 ⊢ ¬ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)
5		psseq1 4043	. . . . 5 ⊢ (𝐴 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵))
6	5	anbi1d 640	. . . 4 ⊢ (𝐴 = 𝐶 → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) ↔ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)))
7	4, 6	mtbiri 329	. . 3 ⊢ (𝐴 = 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶))
8	7	con2i 139	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → ¬ 𝐴 = 𝐶)
9		dfpss2 4041	. 2 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
10	3, 8, 9	sylanbrc 592	1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ⊆ wss 3904   ⊊ wpss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-cleq 2753  df-ne 2957  df-ss 3921  df-pss 3924

This theorem is referenced by:  sspsstr  4062  psssstr  4063  psstrd  4064  porpss  7706  inf3lem5  9584  ltsopr  10987