Theorem bdcuni 10667

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.

Step	Hyp	Ref	Expression
1		ax-bdel 10612	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdex 10610	. . . 4 ⊢ BOUNDED ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3	2	bdcab 10640	. . 3 ⊢ BOUNDED {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧}
4		df-rex 2354	. . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧))
5		exancom 1539	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
6	4, 5	bitri 182	. . . 4 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
7	6	abbii 2194	. . 3 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
8	3, 7	bdceqi 10634	. 2 ⊢ BOUNDED {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
9		df-uni 3602	. 2 ⊢ ∪ 𝑥 = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
10	8, 9	bdceqir 10635	1 ⊢ BOUNDED ∪ 𝑥

Syntax hints:   ∧ wa 102  ∃wex 1421  {cab 2067  ∃wrex 2349  ∪ cuni 3601  BOUNDED wbdc 10631

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdex 10610  ax-bdel 10612  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-rex 2354  df-uni 3602  df-bdc 10632

This theorem is referenced by:  bj-uniex2  10707