Theorem dfuni2 3704

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 3703	. 2 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
2		exancom 1570	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝑦))
3		df-rex 2396	. . . 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 2230	. 2 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
6	1, 5	eqtri 2135	1 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Syntax hints:   ∧ wa 103   = wceq 1314  ∃wex 1451   ∈ wcel 1463  {cab 2101  ∃wrex 2391  ∪ cuni 3702

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-rex 2396  df-uni 3703

This theorem is referenced by:  nfuni  3708  nfunid  3709  unieq  3711  uniiun  3832