Theorem eluniab 3904

Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
eluniab	⊢ (A ∈ ∪{x ∣ φ} ↔ ∃x(A ∈ x ∧ φ))

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem eluniab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3895	. 2 ⊢ (A ∈ ∪{x ∣ φ} ↔ ∃y(A ∈ y ∧ y ∈ {x ∣ φ}))
2		nfv 1619	. . . 4 ⊢ Ⅎx A ∈ y
3		nfsab1 2343	. . . 4 ⊢ Ⅎx y ∈ {x ∣ φ}
4	2, 3	nfan 1824	. . 3 ⊢ Ⅎx(A ∈ y ∧ y ∈ {x ∣ φ})
5		nfv 1619	. . 3 ⊢ Ⅎy(A ∈ x ∧ φ)
6		eleq2 2414	. . . 4 ⊢ (y = x → (A ∈ y ↔ A ∈ x))
7		eleq1 2413	. . . . 5 ⊢ (y = x → (y ∈ {x ∣ φ} ↔ x ∈ {x ∣ φ}))
8		abid 2341	. . . . 5 ⊢ (x ∈ {x ∣ φ} ↔ φ)
9	7, 8	syl6bb 252	. . . 4 ⊢ (y = x → (y ∈ {x ∣ φ} ↔ φ))
10	6, 9	anbi12d 691	. . 3 ⊢ (y = x → ((A ∈ y ∧ y ∈ {x ∣ φ}) ↔ (A ∈ x ∧ φ)))
11	4, 5, 10	cbvex 1985	. 2 ⊢ (∃y(A ∈ y ∧ y ∈ {x ∣ φ}) ↔ ∃x(A ∈ x ∧ φ))
12	1, 11	bitri 240	1 ⊢ (A ∈ ∪{x ∣ φ} ↔ ∃x(A ∈ x ∧ φ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∪cuni 3892

This proof depends on 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 proof 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 used by:  elunirab  3905  dfiun2g  4000  eqtfinrelk  4487  elfv  5327  funiunfv  5468  tcfnex  6245