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

Theorem sssucid 4233
Description: A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
Assertion
Ref Expression
sssucid 𝐴 ⊆ suc 𝐴

Proof of Theorem sssucid
StepHypRef Expression
1 ssun1 3161 . 2 𝐴 ⊆ (𝐴 ∪ {𝐴})
2 df-suc 4189 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
31, 2sseqtr4i 3057 1 𝐴 ⊆ suc 𝐴
Colors of variables: wff set class
Syntax hints:  cun 2995  wss 2997  {csn 3441  suc csuc 4183
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-suc 4189
This theorem is referenced by:  trsuc  4240  ordsuc  4369  0elnn  4422  sucinc  6188  sucinc2  6189  oasuc  6207  phplem4  6551  phplem4dom  6558  phplem4on  6563  fidcenumlemrk  6642  fidcenumlemr  6643  bj-nntrans  11492
  Copyright terms: Public domain W3C validator