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

Theorem bdvsn 15311
Description: Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Assertion
Ref Expression
bdvsn BOUNDED 𝑥 = {𝑦}
Distinct variable group:   𝑥,𝑦

Proof of Theorem bdvsn
StepHypRef Expression
1 bdcsn 15307 . . . 4 BOUNDED {𝑦}
21bdss 15301 . . 3 BOUNDED 𝑥 ⊆ {𝑦}
3 bdcv 15285 . . . 4 BOUNDED 𝑥
43bdsnss 15310 . . 3 BOUNDED {𝑦} ⊆ 𝑥
52, 4ax-bdan 15252 . 2 BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6 eqss 3194 . 2 (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
75, 6bd0r 15262 1 BOUNDED 𝑥 = {𝑦}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wss 3153  {csn 3618  BOUNDED wbd 15249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15250  ax-bdan 15252  ax-bdal 15255  ax-bdeq 15257  ax-bdel 15258  ax-bdsb 15259
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3159  df-ss 3166  df-sn 3624  df-bdc 15278
This theorem is referenced by:  bdop  15312
  Copyright terms: Public domain W3C validator