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Theorem sstr2 3015
 Description: Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
sstr2 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))

Proof of Theorem sstr2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3002 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21imim1d 74 . . 3 (𝐴𝐵 → ((𝑥𝐵𝑥𝐶) → (𝑥𝐴𝑥𝐶)))
32alimdv 1802 . 2 (𝐴𝐵 → (∀𝑥(𝑥𝐵𝑥𝐶) → ∀𝑥(𝑥𝐴𝑥𝐶)))
4 dfss2 2997 . 2 (𝐵𝐶 ↔ ∀𝑥(𝑥𝐵𝑥𝐶))
5 dfss2 2997 . 2 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
63, 4, 53imtr4g 203 1 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1283   ∈ wcel 1434   ⊆ wss 2982 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2988  df-ss 2995 This theorem is referenced by:  sstr  3016  sstri  3017  sseq1  3029  sseq2  3030  ssun3  3147  ssun4  3148  ssinss1  3210  ssdisj  3316  triun  3908  trintssm  3911  sspwb  3999  exss  4010  relss  4473  funss  4970  funimass2  5028  fss  5105  bj-nntrans  11031
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