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Theorem bdvsn 11720
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 11716 . . . 4 BOUNDED {𝑦}
21bdss 11710 . . 3 BOUNDED 𝑥 ⊆ {𝑦}
3 bdcv 11694 . . . 4 BOUNDED 𝑥
43bdsnss 11719 . . 3 BOUNDED {𝑦} ⊆ 𝑥
52, 4ax-bdan 11661 . 2 BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6 eqss 3040 . 2 (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
75, 6bd0r 11671 1 BOUNDED 𝑥 = {𝑦}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wss 2999  {csn 3446  BOUNDED wbd 11658
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-bd0 11659  ax-bdan 11661  ax-bdal 11664  ax-bdeq 11666  ax-bdel 11667  ax-bdsb 11668
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-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-sn 3452  df-bdc 11687
This theorem is referenced by:  bdop  11721
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