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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-uniex | GIF version |
Description: uniex 4455 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 3833 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
3 | 2 | eleq1d 2258 | . 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V)) |
4 | bj-uniex2 15146 | . . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥 | |
5 | 4 | issetri 2761 | . 2 ⊢ ∪ 𝑥 ∈ V |
6 | 1, 3, 5 | vtocl 2806 | 1 ⊢ ∪ 𝐴 ∈ V |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∪ cuni 3824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-un 4451 ax-bd0 15043 ax-bdex 15049 ax-bdel 15051 ax-bdsb 15052 ax-bdsep 15114 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-uni 3825 df-bdc 15071 |
This theorem is referenced by: bj-uniexg 15148 bj-unex 15149 |
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