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Theorem bdcuni 13548
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 13493 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdex 13491 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
32bdcab 13521 . . 3 BOUNDED {𝑦 ∣ ∃𝑧𝑥 𝑦𝑧}
4 df-rex 2441 . . . . 5 (∃𝑧𝑥 𝑦𝑧 ↔ ∃𝑧(𝑧𝑥𝑦𝑧))
5 exancom 1588 . . . . 5 (∃𝑧(𝑧𝑥𝑦𝑧) ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
64, 5bitri 183 . . . 4 (∃𝑧𝑥 𝑦𝑧 ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
76abbii 2273 . . 3 {𝑦 ∣ ∃𝑧𝑥 𝑦𝑧} = {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
83, 7bdceqi 13515 . 2 BOUNDED {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
9 df-uni 3775 . 2 𝑥 = {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
108, 9bdceqir 13516 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wa 103  wex 1472  {cab 2143  wrex 2436   cuni 3774  BOUNDED wbdc 13512
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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-bd0 13485  ax-bdex 13491  ax-bdel 13493  ax-bdsb 13494
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-rex 2441  df-uni 3775  df-bdc 13513
This theorem is referenced by:  bj-uniex2  13588
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