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Mirrors > Home > NFE Home > Th. List > eluni2 | GIF version |
Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.) |
Ref | Expression |
---|---|
eluni2 | ⊢ (A ∈ ∪B ↔ ∃x ∈ B A ∈ x) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1586 | . 2 ⊢ (∃x(A ∈ x ∧ x ∈ B) ↔ ∃x(x ∈ B ∧ A ∈ x)) | |
2 | eluni 3894 | . 2 ⊢ (A ∈ ∪B ↔ ∃x(A ∈ x ∧ x ∈ B)) | |
3 | df-rex 2620 | . 2 ⊢ (∃x ∈ B A ∈ x ↔ ∃x(x ∈ B ∧ A ∈ x)) | |
4 | 1, 2, 3 | 3bitr4i 268 | 1 ⊢ (A ∈ ∪B ↔ ∃x ∈ B A ∈ x) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 ∈ wcel 1710 ∃wrex 2615 ∪cuni 3891 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-rex 2620 df-v 2861 df-uni 3892 |
This theorem is referenced by: uni0b 3916 intssuni 3948 iuncom4 3976 dfuni3 4315 eqpw1uni 4330 elfin 4420 cnvuni 4895 chfnrn 5399 |
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