Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdssex GIF version

Theorem bdssex 11439
Description: Bounded version of ssex 3968. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdssex.bd BOUNDED 𝐴
bdssex.1 𝐵 ∈ V
Assertion
Ref Expression
bdssex (𝐴𝐵𝐴 ∈ V)

Proof of Theorem bdssex
StepHypRef Expression
1 df-ss 3010 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 bdssex.bd . . . 4 BOUNDED 𝐴
3 bdssex.1 . . . 4 𝐵 ∈ V
42, 3bdinex2 11437 . . 3 (𝐴𝐵) ∈ V
5 eleq1 2150 . . 3 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∈ V ↔ 𝐴 ∈ V))
64, 5mpbii 146 . 2 ((𝐴𝐵) = 𝐴𝐴 ∈ V)
71, 6sylbi 119 1 (𝐴𝐵𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  Vcvv 2619  cin 2996  wss 2997  BOUNDED wbdc 11377
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  ax-bdsep 11421
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-in 3003  df-ss 3010  df-bdc 11378
This theorem is referenced by:  bdssexi  11440  bdssexg  11441  bdfind  11487
  Copyright terms: Public domain W3C validator