Theorem uniel 43645

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 2881	. 2 ⊢ ({𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
2		dfuni2 4852	. . 3 ⊢ ∪ 𝐴 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦}
3	2	eleq1i 2827	. 2 ⊢ (∪ 𝐴 ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵)
4		df-rex 3062	. 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 1540  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∃wrex 3061  ∪ cuni 4850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3062  df-uni 4851

This theorem is referenced by:  unielss  43646  onsupmaxb  43667