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Mirrors > Home > NFE Home > Th. List > eluniab | GIF version |
Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
eluniab | ⊢ (A ∈ ∪{x ∣ φ} ↔ ∃x(A ∈ x ∧ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 3894 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 ∪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: elunirab 3904 dfiun2g 3999 eqtfinrelk 4486 elfv 5326 funiunfv 5467 tcfnex 6244 |
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