Theorem dfuni2 3813

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 3812	. 2 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
2		exancom 1608	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝑦))
3		df-rex 2461	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝑦))
4	2, 3	bitr4i 187	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	abbii 2293	. 2 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
6	1, 5	eqtri 2198	1 ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  {cab 2163  ∃wrex 2456  ∪ cuni 3811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-rex 2461  df-uni 3812

This theorem is referenced by:  nfuni  3817  nfunid  3818  unieq  3820  uniiun  3942