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Theorem bdcuni 12908
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 12853 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdex 12851 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
32bdcab 12881 . . 3 BOUNDED {𝑦 ∣ ∃𝑧𝑥 𝑦𝑧}
4 df-rex 2397 . . . . 5 (∃𝑧𝑥 𝑦𝑧 ↔ ∃𝑧(𝑧𝑥𝑦𝑧))
5 exancom 1570 . . . . 5 (∃𝑧(𝑧𝑥𝑦𝑧) ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
64, 5bitri 183 . . . 4 (∃𝑧𝑥 𝑦𝑧 ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
76abbii 2231 . . 3 {𝑦 ∣ ∃𝑧𝑥 𝑦𝑧} = {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
83, 7bdceqi 12875 . 2 BOUNDED {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
9 df-uni 3705 . 2 𝑥 = {𝑦 ∣ ∃𝑧(𝑦𝑧𝑧𝑥)}
108, 9bdceqir 12876 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wa 103  wex 1451  {cab 2101  wrex 2392   cuni 3704  BOUNDED wbdc 12872
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12845  ax-bdex 12851  ax-bdel 12853  ax-bdsb 12854
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-rex 2397  df-uni 3705  df-bdc 12873
This theorem is referenced by:  bj-uniex2  12948
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