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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-uniexg | GIF version |
Description: uniexg 4422 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-uniexg | ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3803 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
2 | 1 | eleq1d 2239 | . 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V)) |
3 | vex 2733 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | bj-uniex 13912 | . 2 ⊢ ∪ 𝑥 ∈ V |
5 | 2, 4 | vtoclg 2790 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ∪ cuni 3794 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-un 4416 ax-bd0 13808 ax-bdex 13814 ax-bdel 13816 ax-bdsb 13817 ax-bdsep 13879 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-uni 3795 df-bdc 13836 |
This theorem is referenced by: (None) |
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