ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sstr2 GIF version

Theorem sstr2 2979
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 2966 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21imim1d 73 . . 3 (𝐴𝐵 → ((𝑥𝐵𝑥𝐶) → (𝑥𝐴𝑥𝐶)))
32alimdv 1775 . 2 (𝐴𝐵 → (∀𝑥(𝑥𝐵𝑥𝐶) → ∀𝑥(𝑥𝐴𝑥𝐶)))
4 dfss2 2961 . 2 (𝐵𝐶 ↔ ∀𝑥(𝑥𝐵𝑥𝐶))
5 dfss2 2961 . 2 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
63, 4, 53imtr4g 198 1 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wcel 1409  wss 2944
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-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-in 2951  df-ss 2958
This theorem is referenced by:  sstr  2980  sstri  2981  sseq1  2993  sseq2  2994  ssun3  3135  ssun4  3136  ssinss1  3192  ssdisj  3304  triun  3894  trintssm  3897  sspwb  3979  exss  3990  relss  4454  funss  4947  funimass2  5004  fss  5081  bj-nntrans  10435
  Copyright terms: Public domain W3C validator