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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-snexg | GIF version |
Description: snexg 4202 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-snexg | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3621 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | bj-prexg 15116 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝐴, 𝐴} ∈ V) | |
3 | 2 | anidms 397 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ∈ V) |
4 | 1, 3 | eqeltrid 2276 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 Vcvv 2752 {csn 3607 {cpr 3608 |
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-14 2163 ax-ext 2171 ax-pr 4227 ax-bdor 15021 ax-bdeq 15025 ax-bdsep 15089 |
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-v 2754 df-un 3148 df-sn 3613 df-pr 3614 |
This theorem is referenced by: bj-snex 15118 bj-sels 15119 bj-sucexg 15127 |
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