Theorem dfuni2 1921

Description: Alternate definition of class union.

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem dfuni2

Step	Hyp	Ref	Expression
1		df-uni 1920	. 2 ⊢ ∪A = {x∣∃y(x ∈ y ∧ y ∈ A)}
2		exancom 736	. . . 4 ⊢ (∃y(x ∈ y ∧ y ∈ A) ↔ ∃y(y ∈ A ∧ x ∈ y))
3		df-rex 1206	. . . 4 ⊢ (∃y ∈ A x ∈ y ↔ ∃y(y ∈ A ∧ x ∈ y))
4	2, 3	bitr4 154	. . 3 ⊢ (∃y(x ∈ y ∧ y ∈ A) ↔ ∃y ∈ A x ∈ y)
5	4	biabi 1181	. 2 ⊢ {x∣∃y(x ∈ y ∧ y ∈ A)} = {x∣∃y ∈ A x ∈ y}
6	1, 5	eqtr 1119	1 ⊢ ∪A = {x∣∃y ∈ A x ∈ y}

Syntax hints:   ∧ wa 196  ∃wex 678   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∪cuni'nbsp;1919

This theorem is referenced by:  uniiun 2026

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-uni 1920

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