Theorem eluni 3895

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 2868	. 2 ⊢ (A ∈ ∪B → A ∈ V)
2		elex 2868	. . . 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 3893	. . 3 ⊢ ∪B = {y ∣ ∃x(y ∈ x ∧ x ∈ B)}
9	7, 8	elab2g 2988	. 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 2860  ∪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-v 2862  df-uni 3893

This theorem is referenced by:  eluni2  3896  elunii  3897  eluniab  3904  unipr  3906  uniun  3911  uniin  3912  uniss  3913  unissb  3922  unipw  4118  eluni1g  4173  unipw1  4326  nnadjoinlem1  4520  dmuni  4915  rnuni  5039  fununi  5161  funiunfv  5468  pw1fnex  5853  tcfnex  6245