Theorem uniel 43669

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 2885	. 2 ⊢ ({𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
2		dfuni2 4847	. . 3 ⊢ ∪ 𝐴 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦}
3	2	eleq1i 2831	. 2 ⊢ (∪ 𝐴 ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵)
4		df-rex 3065	. 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 304	1 ⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786   ∈ wcel 2119  {cab 2718  ∃wrex 3064  ∪ cuni 4845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065  df-uni 4846

This theorem is referenced by:  unielss  43670  onsupmaxb  43691