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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-snexg | GIF version |
Description: snexg 3936 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 3389 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | bj-prexg 10031 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝐴, 𝐴} ∈ V) | |
3 | 2 | anidms 377 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ∈ V) |
4 | 1, 3 | syl5eqel 2124 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 Vcvv 2557 {csn 3375 {cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-pr 3944 ax-bdor 9936 ax-bdeq 9940 ax-bdsep 10004 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 |
This theorem is referenced by: bj-snex 10033 bj-sels 10034 bj-sucexg 10042 |
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