Theorem eluni2 3575

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 1496	. 2 ⊢ (∃x(A ∈ x ∧ x ∈ B) ↔ ∃x(x ∈ B ∧ A ∈ x))
2		eluni 3574	. 2 ⊢ (A ∈ ∪ B ↔ ∃x(A ∈ x ∧ x ∈ B))
3		df-rex 2306	. 2 ⊢ (∃x ∈ B A ∈ x ↔ ∃x(x ∈ B ∧ A ∈ x))
4	1, 2, 3	3bitr4i 201	1 ⊢ (A ∈ ∪ B ↔ ∃x ∈ B A ∈ x)

Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ∪ cuni 3571

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-uni 3572

This theorem is referenced by:  uni0b  3596  intssunim  3628  iuncom4  3655  inuni  3900  ssorduni  4179  unon  4202  cnvuni  4464  chfnrn  5221