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Theorem psssstr 4013
 Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
psssstr ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem psssstr
StepHypRef Expression
1 sspss 4006 . 2 (𝐵𝐶 ↔ (𝐵𝐶𝐵 = 𝐶))
2 psstr 4011 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 417 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq2 3995 . . . . 5 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
54biimpcd 252 . . . 4 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
63, 5jaod 857 . . 3 (𝐴𝐵 → ((𝐵𝐶𝐵 = 𝐶) → 𝐴𝐶))
76imp 411 . 2 ((𝐴𝐵 ∧ (𝐵𝐶𝐵 = 𝐶)) → 𝐴𝐶)
81, 7sylan2b 597 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 845   = wceq 1539   ⊆ wss 3859   ⊊ wpss 3860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-v 3412  df-in 3866  df-ss 3876  df-pss 3878 This theorem is referenced by:  psssstrd  4016  suplem1pr  10502  atexch  30253  bj-2upln0  34730  bj-2upln1upl  34731
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