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Mirrors > Home > ILE Home > Th. List > uniex2 | GIF version |
Description: The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.) |
Ref | Expression |
---|---|
uniex2 | ⊢ ∃𝑦 𝑦 = ∪ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfun 4171 | . . . 4 ⊢ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
2 | eluni 3583 | . . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)) | |
3 | 2 | imbi1i 227 | . . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)) |
4 | 3 | albii 1359 | . . . . 5 ⊢ (∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)) |
5 | 4 | exbii 1496 | . . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)) |
6 | 1, 5 | mpbir 134 | . . 3 ⊢ ∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) |
7 | 6 | bm1.3ii 3878 | . 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥) |
8 | dfcleq 2034 | . . 3 ⊢ (𝑦 = ∪ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥)) | |
9 | 8 | exbii 1496 | . 2 ⊢ (∃𝑦 𝑦 = ∪ 𝑥 ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥)) |
10 | 7, 9 | mpbir 134 | 1 ⊢ ∃𝑦 𝑦 = ∪ 𝑥 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∪ cuni 3580 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-uni 3581 |
This theorem is referenced by: uniex 4174 |
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