Theorem dfuni2 3894

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 3893	. 2 ⊢ ∪A = {x ∣ ∃y(x ∈ y ∧ y ∈ A)}
2		exancom 1586	. . . 4 ⊢ (∃y(x ∈ y ∧ y ∈ A) ↔ ∃y(y ∈ A ∧ x ∈ y))
3		df-rex 2621	. . . 4 ⊢ (∃y ∈ A x ∈ y ↔ ∃y(y ∈ A ∧ x ∈ y))
4	2, 3	bitr4i 243	. . 3 ⊢ (∃y(x ∈ y ∧ y ∈ A) ↔ ∃y ∈ A x ∈ y)
5	4	abbii 2466	. 2 ⊢ {x ∣ ∃y(x ∈ y ∧ y ∈ A)} = {x ∣ ∃y ∈ A x ∈ y}
6	1, 5	eqtri 2373	1 ⊢ ∪A = {x ∣ ∃y ∈ A x ∈ y}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-rex 2621  df-uni 3893

This theorem is referenced by:  nfuni  3898  nfunid  3899  unieq  3901  uniiun  4020