Theorem sspsstr 4104

Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Assertion

Ref	Expression
sspsstr	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)

Proof of Theorem sspsstr

Step	Hyp	Ref	Expression
1		sspss 4098	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
2		psstr 4103	. . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
3	2	ex 411	. . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶))
4		psseq1 4086	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
5	4	biimprd 247	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶))
6	3, 5	jaoi 855	. . 3 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶))
7	6	imp 405	. 2 ⊢ (((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
8	1, 7	sylanb 579	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 845   = wceq 1534   ⊆ wss 3947   ⊊ wpss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-cleq 2718  df-ne 2931  df-ss 3964  df-pss 3967

This theorem is referenced by:  sspsstrd  4107  ordtr2  6420  php  9244  phpOLD  9256  canthp1lem2  10696  suplem1pr  11095  fbfinnfr  23836  ppiltx  27205