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Theorem bdcuni 11424
Description: The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
Assertion
Ref Expression
bdcuni BOUNDED 𝑥

Proof of Theorem bdcuni
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-bdel 11369 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdex 11367 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
32bdcab 11397 . . 3 BOUNDED {𝑦 ∣ ∃𝑧𝑥 𝑦𝑧}
4 df-rex 2365 . . . . 5 (∃𝑧𝑥 𝑦𝑧 ↔ ∃𝑧(𝑧𝑥𝑦𝑧))
5 exancom 1544 . . . . 5 (∃𝑧(𝑧𝑥𝑦𝑧) ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
64, 5bitri 182 . . . 4 (∃𝑧𝑥 𝑦𝑧 ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
76abbii 2203 . . 3 {𝑦 ∣ ∃𝑧𝑥 𝑦𝑧} = {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
83, 7bdceqi 11391 . 2 BOUNDED {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
9 df-uni 3649 . 2 𝑥 = {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
108, 9bdceqir 11392 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wa 102  wex 1426  {cab 2074  wrex 2360   cuni 3648  BOUNDED wbdc 11388
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11361  ax-bdex 11367  ax-bdel 11369  ax-bdsb 11370
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-rex 2365  df-uni 3649  df-bdc 11389
This theorem is referenced by:  bj-uniex2  11464
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