Theorem uniel 43249

Description: Two ways to say a union is an element of a class. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
uniel	⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵

Allowed substitution hints:   𝐵(𝑦,𝑧)

Proof of Theorem uniel

Step	Hyp	Ref	Expression
1		clabel 2877	. 2 ⊢ ({𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
2		dfuni2 4861	. . 3 ⊢ ∪ 𝐴 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦}
3	2	eleq1i 2822	. 2 ⊢ (∪ 𝐴 ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵)
4		df-rex 3057	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
5	1, 3, 4	3bitr4i 303	1 ⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1539  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∃wrex 3056  ∪ cuni 4859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-uni 4860

This theorem is referenced by:  unielss  43250  onsupmaxb  43271