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Theorem psssstr 4080
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 4073 . 2 (𝐵𝐶 ↔ (𝐵𝐶𝐵 = 𝐶))
2 psstr 4078 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 413 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq2 4062 . . . . 5 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
54biimpcd 250 . . . 4 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
63, 5jaod 853 . . 3 (𝐴𝐵 → ((𝐵𝐶𝐵 = 𝐶) → 𝐴𝐶))
76imp 407 . 2 ((𝐴𝐵 ∧ (𝐵𝐶𝐵 = 𝐶)) → 𝐴𝐶)
81, 7sylan2b 593 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wss 3933  wpss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-in 3940  df-ss 3949  df-pss 3951
This theorem is referenced by:  psssstrd  4083  suplem1pr  10462  atexch  30085  bj-2upln0  34232  bj-2upln1upl  34233
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