Theorem eluniab 1926

Description: Membership in union of a class abstract.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem eluniab

Step	Hyp	Ref	Expression
1		eluni 1922	. 2 ⊢ (A ∈ ∪{x∣φ} ↔ ∃y(A ∈ y ∧ y ∈ {x∣φ}))
2		ax-17 925	. . . 4 ⊢ (A ∈ y → ∀x A ∈ y)
3		hbab1 1095	. . . 4 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
4	2, 3	hban 704	. . 3 ⊢ ((A ∈ y ∧ y ∈ {x∣φ}) → ∀x(A ∈ y ∧ y ∈ {x∣φ}))
5		ax-17 925	. . 3 ⊢ ((A ∈ x ∧ φ) → ∀y(A ∈ x ∧ φ))
6		eleq2 1150	. . . . 5 ⊢ (y = x → (A ∈ y ↔ A ∈ x))
7		eleq1 1149	. . . . 5 ⊢ (y = x → (y ∈ {x∣φ} ↔ x ∈ {x∣φ}))
8	6, 7	anbi12d 476	. . . 4 ⊢ (y = x → ((A ∈ y ∧ y ∈ {x∣φ}) ↔ (A ∈ x ∧ x ∈ {x∣φ})))
9		abid 1094	. . . . 5 ⊢ (x ∈ {x∣φ} ↔ φ)
10	9	anbi2i 367	. . . 4 ⊢ ((A ∈ x ∧ x ∈ {x∣φ}) ↔ (A ∈ x ∧ φ))
11	8, 10	syl6bb 414	. . 3 ⊢ (y = x → ((A ∈ y ∧ y ∈ {x∣φ}) ↔ (A ∈ x ∧ φ)))
12	4, 5, 11	cbvex 849	. 2 ⊢ (∃y(A ∈ y ∧ y ∈ {x∣φ}) ↔ ∃x(A ∈ x ∧ φ))
13	1, 12	bitr 151	1 ⊢ (A ∈ ∪{x∣φ} ↔ ∃x(A ∈ x ∧ φ))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   ∈ wcel 1092  ∪cuni 1919

This theorem is referenced by:  elfv 2830  tfrlem8 2956  tfrlem9 2957  aceq5lem2 3559

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-uni 1920

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