Theorem dfuni2 4841

Description: Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
dfuni2	⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfuni2

Step	Hyp	Ref	Expression
1		df-uni 4840	. 2 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
2		exancom 1864	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝑦))
3		df-rex 3070	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝑦))
4	2, 3	bitr4i 277	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	abbii 2808	. 2 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
6	1, 5	eqtri 2766	1 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Syntax hints:   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∃wrex 3065  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-rex 3070  df-uni 4840

This theorem is referenced by:  nfunid  4845  unieqOLD  4851  uniiun  4988  rncnvepres  36439