Theorem uniel 44480

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 2909	. 2 ⊢ ({𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
2		dfuni2 4874	. . 3 ⊢ ∪ 𝐴 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦}
3	2	eleq1i 2855	. 2 ⊢ (∪ 𝐴 ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} ∈ 𝐵)
4		df-rex 3091	. 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 306	1 ⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090  ∪ cuni 4872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-uni 4873

This theorem is used by:  unielss  44481  onsupmaxb  44502