Theorem dfuni2 3791

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 3790	. 2 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
2		exancom 1596	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝑦))
3		df-rex 2450	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝑦))
4	2, 3	bitr4i 186	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	abbii 2282	. 2 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
6	1, 5	eqtri 2186	1 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  ∃wrex 2445  ∪ cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-rex 2450  df-uni 3790

This theorem is referenced by:  nfuni  3795  nfunid  3796  unieq  3798  uniiun  3919