Theorem bdcuni 15950

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 15895	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdex 15893	. . . 4 ⊢ BOUNDED ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3	2	bdcab 15923	. . 3 ⊢ BOUNDED {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧}
4		df-rex 2491	. . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧))
5		exancom 1632	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
6	4, 5	bitri 184	. . . 4 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
7	6	abbii 2322	. . 3 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
8	3, 7	bdceqi 15917	. 2 ⊢ BOUNDED {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
9		df-uni 3857	. 2 ⊢ ∪ 𝑥 = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
10	8, 9	bdceqir 15918	1 ⊢ BOUNDED ∪ 𝑥

Syntax hints:   ∧ wa 104  ∃wex 1516  {cab 2192  ∃wrex 2486  ∪ cuni 3856  BOUNDED wbdc 15914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-bd0 15887  ax-bdex 15893  ax-bdel 15895  ax-bdsb 15896

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-rex 2491  df-uni 3857  df-bdc 15915

This theorem is referenced by:  bj-uniex2  15990