Theorem eluni 3894

Description: Membership in class union. (Contributed by NM, 22-May-1994.)

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem eluni

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2867	. 2 ⊢ (A ∈ ∪B → A ∈ V)
2		elex 2867	. . . 4 ⊢ (A ∈ x → A ∈ V)
3	2	adantr 451	. . 3 ⊢ ((A ∈ x ∧ x ∈ B) → A ∈ V)
4	3	exlimiv 1634	. 2 ⊢ (∃x(A ∈ x ∧ x ∈ B) → A ∈ V)
5		eleq1 2413	. . . . 5 ⊢ (y = A → (y ∈ x ↔ A ∈ x))
6	5	anbi1d 685	. . . 4 ⊢ (y = A → ((y ∈ x ∧ x ∈ B) ↔ (A ∈ x ∧ x ∈ B)))
7	6	exbidv 1626	. . 3 ⊢ (y = A → (∃x(y ∈ x ∧ x ∈ B) ↔ ∃x(A ∈ x ∧ x ∈ B)))
8		df-uni 3892	. . 3 ⊢ ∪B = {y ∣ ∃x(y ∈ x ∧ x ∈ B)}
9	7, 8	elab2g 2987	. 2 ⊢ (A ∈ V → (A ∈ ∪B ↔ ∃x(A ∈ x ∧ x ∈ B)))
10	1, 4, 9	pm5.21nii 342	1 ⊢ (A ∈ ∪B ↔ ∃x(A ∈ x ∧ x ∈ B))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ∪cuni 3891

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 2478  df-v 2861  df-uni 3892

This theorem is referenced by:  eluni2  3895  elunii  3896  eluniab  3903  unipr  3905  uniun  3910  uniin  3911  uniss  3912  unissb  3921  unipw  4117  eluni1g  4172  unipw1  4325  nnadjoinlem1  4519  dmuni  4914  rnuni  5038  fununi  5160  funiunfv  5467  pw1fnex  5852  tcfnex  6244