Theorem eluni2 3896

Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)

Assertion

Ref	Expression
eluni2	⊢ (A ∈ ∪B ↔ ∃x ∈ B A ∈ x)

Distinct variable groups:   x,A   x,B

Proof of Theorem eluni2

Step	Hyp	Ref	Expression
1		exancom 1586	. 2 ⊢ (∃x(A ∈ x ∧ x ∈ B) ↔ ∃x(x ∈ B ∧ A ∈ x))
2		eluni 3895	. 2 ⊢ (A ∈ ∪B ↔ ∃x(A ∈ x ∧ x ∈ B))
3		df-rex 2621	. 2 ⊢ (∃x ∈ B A ∈ x ↔ ∃x(x ∈ B ∧ A ∈ x))
4	1, 2, 3	3bitr4i 268	1 ⊢ (A ∈ ∪B ↔ ∃x ∈ B A ∈ x)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862  df-uni 3893

This theorem is referenced by:  uni0b  3917  intssuni  3949  iuncom4  3977  dfuni3  4316  eqpw1uni  4331  elfin  4421  cnvuni  4896  chfnrn  5400