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Theorem psssstr 3051
 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 pssss 3039 . . 3 (𝐴𝐵𝐴𝐵)
2 id 19 . . 3 (𝐵𝐶𝐵𝐶)
31, 2sylan9ss 2958 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 sspssn 3048 . . . . 5 ¬ (𝐵𝐶𝐶𝐵)
5 psseq1 3031 . . . . . 6 (𝐴 = 𝐶 → (𝐴𝐵𝐶𝐵))
65anbi2d 437 . . . . 5 (𝐴 = 𝐶 → ((𝐵𝐶𝐴𝐵) ↔ (𝐵𝐶𝐶𝐵)))
74, 6mtbiri 600 . . . 4 (𝐴 = 𝐶 → ¬ (𝐵𝐶𝐴𝐵))
87con2i 557 . . 3 ((𝐵𝐶𝐴𝐵) → ¬ 𝐴 = 𝐶)
98ancoms 255 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 = 𝐶)
10 dfpss2 3029 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
113, 9, 10sylanbrc 394 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243   ⊆ wss 2917   ⊊ wpss 2918 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933 This theorem is referenced by:  psssstrd  3054
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