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Theorem sstr2 2925
 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 2912 . . . 4 (AB → (x Ax B))
21imim1d 69 . . 3 (AB → ((x Bx 𝐶) → (x Ax 𝐶)))
32alimdv 1737 . 2 (AB → (x(x Bx 𝐶) → x(x Ax 𝐶)))
4 dfss2 2907 . 2 (B𝐶x(x Bx 𝐶))
5 dfss2 2907 . 2 (A𝐶x(x Ax 𝐶))
63, 4, 53imtr4g 194 1 (AB → (B𝐶A𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1224   ∈ wcel 1370   ⊆ wss 2890 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-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904 This theorem is referenced by:  sstr  2926  sstri  2927  sseq1  2939  sseq2  2940  ssun3  3081  ssun4  3082  ssinss1  3138  ssdisj  3250  triun  3837  trint0m  3841  sspwb  3922  exss  3933  relss  4350  funss  4842  funimass2  4899  fss  4976  bj-nntrans  7369
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