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Theorem bdvsn 13156
 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 13152 . . . 4 BOUNDED {𝑦}
21bdss 13146 . . 3 BOUNDED 𝑥 ⊆ {𝑦}
3 bdcv 13130 . . . 4 BOUNDED 𝑥
43bdsnss 13155 . . 3 BOUNDED {𝑦} ⊆ 𝑥
52, 4ax-bdan 13097 . 2 BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6 eqss 3112 . 2 (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
75, 6bd0r 13107 1 BOUNDED 𝑥 = {𝑦}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1331   ⊆ wss 3071  {csn 3527  BOUNDED wbd 13094 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13095  ax-bdan 13097  ax-bdal 13100  ax-bdeq 13102  ax-bdel 13103  ax-bdsb 13104 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533  df-bdc 13123 This theorem is referenced by:  bdop  13157
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