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Theorem sspsstr 3374
 Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
sspsstr ((A B BC) → AC)

Proof of Theorem sspsstr
StepHypRef Expression
1 sspss 3368 . 2 (A B ↔ (AB A = B))
2 psstr 3373 . . . . 5 ((AB BC) → AC)
32ex 423 . . . 4 (AB → (BCAC))
4 psseq1 3356 . . . . 5 (A = B → (ACBC))
54biimprd 214 . . . 4 (A = B → (BCAC))
63, 5jaoi 368 . . 3 ((AB A = B) → (BCAC))
76imp 418 . 2 (((AB A = B) BC) → AC)
81, 7sylanb 458 1 ((A B BC) → AC)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by:  sspsstrd  3377
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