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Theorem bdvsn 15079
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 15075 . . . 4 BOUNDED {𝑦}
21bdss 15069 . . 3 BOUNDED 𝑥 ⊆ {𝑦}
3 bdcv 15053 . . . 4 BOUNDED 𝑥
43bdsnss 15078 . . 3 BOUNDED {𝑦} ⊆ 𝑥
52, 4ax-bdan 15020 . 2 BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6 eqss 3185 . 2 (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
75, 6bd0r 15030 1 BOUNDED 𝑥 = {𝑦}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wss 3144  {csn 3607  BOUNDED wbd 15017
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 2171  ax-bd0 15018  ax-bdan 15020  ax-bdal 15023  ax-bdeq 15025  ax-bdel 15026  ax-bdsb 15027
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-sn 3613  df-bdc 15046
This theorem is referenced by:  bdop  15080
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