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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-uniex | GIF version |
Description: uniex 4367 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 3753 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
3 | 2 | eleq1d 2209 | . 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V)) |
4 | bj-uniex2 13285 | . . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥 | |
5 | 4 | issetri 2698 | . 2 ⊢ ∪ 𝑥 ∈ V |
6 | 1, 3, 5 | vtocl 2743 | 1 ⊢ ∪ 𝐴 ∈ V |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∪ cuni 3744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-un 4363 ax-bd0 13182 ax-bdex 13188 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-uni 3745 df-bdc 13210 |
This theorem is referenced by: bj-uniexg 13287 bj-unex 13288 |
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