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Theorem sspsstr 3044
 Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
sspsstr ((AB B𝐶) → A𝐶)

Proof of Theorem sspsstr
StepHypRef Expression
1 id 19 . . 3 (ABAB)
2 pssss 3033 . . 3 (B𝐶B𝐶)
31, 2sylan9ss 2952 . 2 ((AB B𝐶) → A𝐶)
4 sspssn 3042 . . . 4 ¬ (𝐶B B𝐶)
5 sseq1 2960 . . . . 5 (A = 𝐶 → (AB𝐶B))
65anbi1d 438 . . . 4 (A = 𝐶 → ((AB B𝐶) ↔ (𝐶B B𝐶)))
74, 6mtbiri 599 . . 3 (A = 𝐶 → ¬ (AB B𝐶))
87con2i 557 . 2 ((AB B𝐶) → ¬ A = 𝐶)
9 dfpss2 3023 . 2 (A𝐶 ↔ (A𝐶 ¬ A = 𝐶))
103, 8, 9sylanbrc 394 1 ((AB B𝐶) → A𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1242   ⊆ wss 2911   ⊊ wpss 2912 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ne 2203  df-in 2918  df-ss 2925  df-pss 2927 This theorem is referenced by:  sspsstrd  3047
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