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Theorem psssstr 3861
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 3854 . 2 (𝐵𝐶 ↔ (𝐵𝐶𝐵 = 𝐶))
2 psstr 3859 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 397 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq2 3843 . . . . 5 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
54biimpcd 239 . . . 4 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
63, 5jaod 839 . . 3 (𝐴𝐵 → ((𝐵𝐶𝐵 = 𝐶) → 𝐴𝐶))
76imp 393 . 2 ((𝐴𝐵 ∧ (𝐵𝐶𝐵 = 𝐶)) → 𝐴𝐶)
81, 7sylan2b 573 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 826   = wceq 1630  wss 3721  wpss 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-ne 2943  df-in 3728  df-ss 3735  df-pss 3737
This theorem is referenced by:  psssstrd  3864  suplem1pr  10075  atexch  29574  bj-2upln0  33336  bj-2upln1upl  33337
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