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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-uniex | GIF version |
Description: uniex 4192 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-uniex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
bj-uniex | ⊢ ∪ 𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-uniex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | unieq 3610 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
3 | 2 | eleq1d 2147 | . 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V)) |
4 | bj-uniex2 10707 | . . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥 | |
5 | 4 | issetri 2608 | . 2 ⊢ ∪ 𝑥 ∈ V |
6 | 1, 3, 5 | vtocl 2653 | 1 ⊢ ∪ 𝐴 ∈ V |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 Vcvv 2601 ∪ cuni 3601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-un 4188 ax-bd0 10604 ax-bdex 10610 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-uni 3602 df-bdc 10632 |
This theorem is referenced by: bj-uniexg 10709 bj-unex 10710 |
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