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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-uniexg | GIF version |
Description: uniexg 4451 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 3830 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
2 | 1 | eleq1d 2256 | . 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V)) |
3 | vex 2752 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | bj-uniex 14940 | . 2 ⊢ ∪ 𝑥 ∈ V |
5 | 2, 4 | vtoclg 2809 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ∪ cuni 3821 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-un 4445 ax-bd0 14836 ax-bdex 14842 ax-bdel 14844 ax-bdsb 14845 ax-bdsep 14907 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-rex 2471 df-v 2751 df-uni 3822 df-bdc 14864 |
This theorem is referenced by: (None) |
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