Theorem psssstr 3713

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

Assertion

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

Proof of Theorem psssstr

Step	Hyp	Ref	Expression
1		sspss 3706	. 2 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶))
2		psstr 3711	. . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
3	2	ex 450	. . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶))
4		psseq2 3695	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ 𝐶))
5	4	biimpcd 239	. . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 = 𝐶 → 𝐴 ⊊ 𝐶))
6	3, 5	jaod 395	. . 3 ⊢ (𝐴 ⊊ 𝐵 → ((𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶) → 𝐴 ⊊ 𝐶))
7	6	imp 445	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶)) → 𝐴 ⊊ 𝐶)
8	1, 7	sylan2b 492	1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ⊆ wss 3574   ⊊ wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  psssstrd  3716  suplem1pr  9874  atexch  29240  bj-2upln0  33011  bj-2upln1upl  33012