Theorem bdcuni 15313

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 15258	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdex 15256	. . . 4 ⊢ BOUNDED ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3	2	bdcab 15286	. . 3 ⊢ BOUNDED {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧}
4		df-rex 2478	. . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧))
5		exancom 1619	. . . . 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 2309	. . 3 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
8	3, 7	bdceqi 15280	. 2 ⊢ BOUNDED {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
9		df-uni 3836	. 2 ⊢ ∪ 𝑥 = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
10	8, 9	bdceqir 15281	1 ⊢ BOUNDED ∪ 𝑥

Syntax hints:   ∧ wa 104  ∃wex 1503  {cab 2179  ∃wrex 2473  ∪ cuni 3835  BOUNDED wbdc 15277

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15250  ax-bdex 15256  ax-bdel 15258  ax-bdsb 15259

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-rex 2478  df-uni 3836  df-bdc 15278

This theorem is referenced by:  bj-uniex2  15353