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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-snexg | GIF version |
Description: snexg 4116 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 3546 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | bj-prexg 13280 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝐴, 𝐴} ∈ V) | |
3 | 2 | anidms 395 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ∈ V) |
4 | 1, 3 | eqeltrid 2227 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 Vcvv 2689 {csn 3532 {cpr 3533 |
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-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-pr 4139 ax-bdor 13185 ax-bdeq 13189 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-v 2691 df-un 3080 df-sn 3538 df-pr 3539 |
This theorem is referenced by: bj-snex 13282 bj-sels 13283 bj-sucexg 13291 |
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