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Mirrors > Home > ILE 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 1496 | . 2 ⊢ (∃x(A ∈ x ∧ x ∈ B) ↔ ∃x(x ∈ B ∧ A ∈ x)) | |
2 | eluni 3574 | . 2 ⊢ (A ∈ ∪ B ↔ ∃x(A ∈ x ∧ x ∈ B)) | |
3 | df-rex 2306 | . 2 ⊢ (∃x ∈ B A ∈ x ↔ ∃x(x ∈ B ∧ A ∈ x)) | |
4 | 1, 2, 3 | 3bitr4i 201 | 1 ⊢ (A ∈ ∪ B ↔ ∃x ∈ B A ∈ x) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 ∪ cuni 3571 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-uni 3572 |
This theorem is referenced by: uni0b 3596 intssunim 3628 iuncom4 3655 inuni 3900 ssorduni 4179 unon 4202 cnvuni 4464 chfnrn 5221 |
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