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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-uniex2 | Structured version GIF version |
Description: uniex2 4121 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-uniex2 | ⊢ ∃y y = ∪ x |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdcuni 6449 | . . . 4 ⊢ BOUNDED ∪ x | |
2 | 1 | bdeli 6421 | . . 3 ⊢ BOUNDED z ∈ ∪ x |
3 | zfun 4119 | . . . 4 ⊢ ∃y∀z(∃y(z ∈ y ∧ y ∈ x) → z ∈ y) | |
4 | eluni 3556 | . . . . . . 7 ⊢ (z ∈ ∪ x ↔ ∃y(z ∈ y ∧ y ∈ x)) | |
5 | 4 | imbi1i 227 | . . . . . 6 ⊢ ((z ∈ ∪ x → z ∈ y) ↔ (∃y(z ∈ y ∧ y ∈ x) → z ∈ y)) |
6 | 5 | albii 1339 | . . . . 5 ⊢ (∀z(z ∈ ∪ x → z ∈ y) ↔ ∀z(∃y(z ∈ y ∧ y ∈ x) → z ∈ y)) |
7 | 6 | exbii 1480 | . . . 4 ⊢ (∃y∀z(z ∈ ∪ x → z ∈ y) ↔ ∃y∀z(∃y(z ∈ y ∧ y ∈ x) → z ∈ y)) |
8 | 3, 7 | mpbir 134 | . . 3 ⊢ ∃y∀z(z ∈ ∪ x → z ∈ y) |
9 | 2, 8 | bdbm1.3ii 6461 | . 2 ⊢ ∃y∀z(z ∈ y ↔ z ∈ ∪ x) |
10 | dfcleq 2017 | . . 3 ⊢ (y = ∪ x ↔ ∀z(z ∈ y ↔ z ∈ ∪ x)) | |
11 | 10 | exbii 1480 | . 2 ⊢ (∃y y = ∪ x ↔ ∃y∀z(z ∈ y ↔ z ∈ ∪ x)) |
12 | 9, 11 | mpbir 134 | 1 ⊢ ∃y y = ∪ x |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1226 = wceq 1228 ∃wex 1363 ∈ wcel 1375 ∪ cuni 3553 |
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 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-8 1377 ax-10 1378 ax-11 1379 ax-i12 1380 ax-bnd 1381 ax-4 1382 ax-13 1386 ax-14 1387 ax-17 1401 ax-i9 1405 ax-ial 1410 ax-i5r 1411 ax-ext 2005 ax-un 4118 ax-bd0 6389 ax-bdex 6395 ax-bdel 6397 ax-bdsb 6398 ax-bdsep 6455 |
This theorem depends on definitions: df-bi 110 df-tru 1231 df-nf 1330 df-sb 1629 df-clab 2010 df-cleq 2016 df-clel 2019 df-nfc 2150 df-rex 2289 df-v 2536 df-uni 3554 df-bdc 6416 |
This theorem is referenced by: bj-uniex 6484 |
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