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Theorem bdvsn 7101
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 7097 . . . 4 BOUNDED {y}
21bdss 7091 . . 3 BOUNDED x ⊆ {y}
3 bdcv 7075 . . . 4 BOUNDED x
43bdsnss 7100 . . 3 BOUNDED {y} ⊆ x
52, 4ax-bdan 7042 . 2 BOUNDED (x ⊆ {y} {y} ⊆ x)
6 eqss 2937 . 2 (x = {y} ↔ (x ⊆ {y} {y} ⊆ x))
75, 6bd0r 7052 1 BOUNDED x = {y}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wss 2894  {csn 3350  BOUNDED wbd 7039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bd0 7040  ax-bdan 7042  ax-bdal 7045  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-sn 3356  df-bdc 7068
This theorem is referenced by:  bdop  7102
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