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Mirrors > Home > ILE Home > Th. List > unisn3 | GIF version |
Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
Ref | Expression |
---|---|
unisn3 | ⊢ (A ∈ B → ∪ {x ∈ B ∣ x = A} = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabsn 3428 | . . 3 ⊢ (A ∈ B → {x ∈ B ∣ x = A} = {A}) | |
2 | 1 | unieqd 3582 | . 2 ⊢ (A ∈ B → ∪ {x ∈ B ∣ x = A} = ∪ {A}) |
3 | unisng 3588 | . 2 ⊢ (A ∈ B → ∪ {A} = A) | |
4 | 2, 3 | eqtrd 2069 | 1 ⊢ (A ∈ B → ∪ {x ∈ B ∣ x = A} = A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 {crab 2304 {csn 3367 ∪ 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-rab 2309 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-uni 3572 |
This theorem is referenced by: (None) |
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