![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-snexg | GIF version |
Description: snexg 4017 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 3458 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | bj-prexg 11685 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝐴, 𝐴} ∈ V) | |
3 | 2 | anidms 389 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ∈ V) |
4 | 1, 3 | syl5eqel 2174 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 Vcvv 2619 {csn 3444 {cpr 3445 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-pr 4034 ax-bdor 11590 ax-bdeq 11594 ax-bdsep 11658 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-sn 3450 df-pr 3451 |
This theorem is referenced by: bj-snex 11687 bj-sels 11688 bj-sucexg 11696 |
Copyright terms: Public domain | W3C validator |