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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 4137 | . . . 4 ⊢ ∃y∀z(∃y(z ∈ y ∧ y ∈ x) → z ∈ y) | |
2 | eluni 3574 | . . . . . . 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 1356 | . . . . 5 ⊢ (∀z(z ∈ ∪ x → z ∈ y) ↔ ∀z(∃y(z ∈ y ∧ y ∈ x) → z ∈ y)) |
5 | 4 | exbii 1493 | . . . 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 3869 | . 2 ⊢ ∃y∀z(z ∈ y ↔ z ∈ ∪ x) |
8 | dfcleq 2031 | . . 3 ⊢ (y = ∪ x ↔ ∀z(z ∈ y ↔ z ∈ ∪ x)) | |
9 | 8 | exbii 1493 | . 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 1240 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∪ 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-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-un 4136 |
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-v 2553 df-uni 3572 |
This theorem is referenced by: uniex 4140 |
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