Theorem bdcuni 15606

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 15551	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdex 15549	. . . 4 ⊢ BOUNDED ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3	2	bdcab 15579	. . 3 ⊢ BOUNDED {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧}
4		df-rex 2481	. . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧))
5		exancom 1622	. . . . 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 2312	. . 3 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
8	3, 7	bdceqi 15573	. 2 ⊢ BOUNDED {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
9		df-uni 3841	. 2 ⊢ ∪ 𝑥 = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)}
10	8, 9	bdceqir 15574	1 ⊢ BOUNDED ∪ 𝑥

Syntax hints:   ∧ wa 104  ∃wex 1506  {cab 2182  ∃wrex 2476  ∪ cuni 3840  BOUNDED wbdc 15570

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15543  ax-bdex 15549  ax-bdel 15551  ax-bdsb 15552

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rex 2481  df-uni 3841  df-bdc 15571

This theorem is referenced by:  bj-uniex2  15646