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Theorem psssstr 3078
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 3066 . . 3 (𝐴𝐵𝐴𝐵)
2 id 19 . . 3 (𝐵𝐶𝐵𝐶)
31, 2sylan9ss 2985 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 sspssn 3075 . . . . 5 ¬ (𝐵𝐶𝐶𝐵)
5 psseq1 3058 . . . . . 6 (𝐴 = 𝐶 → (𝐴𝐵𝐶𝐵))
65anbi2d 445 . . . . 5 (𝐴 = 𝐶 → ((𝐵𝐶𝐴𝐵) ↔ (𝐵𝐶𝐶𝐵)))
74, 6mtbiri 610 . . . 4 (𝐴 = 𝐶 → ¬ (𝐵𝐶𝐴𝐵))
87con2i 567 . . 3 ((𝐵𝐶𝐴𝐵) → ¬ 𝐴 = 𝐶)
98ancoms 259 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 = 𝐶)
10 dfpss2 3056 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
113, 9, 10sylanbrc 402 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101   = wceq 1259  wss 2944  wpss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ne 2221  df-in 2951  df-ss 2958  df-pss 2960
This theorem is referenced by:  psssstrd  3081
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