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Mirrors > Home > ILE Home > Th. List > ssexi | GIF version |
Description: The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
Ref | Expression |
---|---|
ssexi.1 | ⊢ 𝐵 ∈ V |
ssexi.2 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
ssexi | ⊢ 𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexi.2 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | ssexi.1 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 2 | ssex 4005 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
4 | 1, 3 | ax-mp 7 | 1 ⊢ 𝐴 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 Vcvv 2641 ⊆ wss 3021 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-in 3027 df-ss 3034 |
This theorem is referenced by: zfausab 4010 pp0ex 4053 ord3ex 4054 epse 4202 opabex 5576 oprabex 5957 phplem2 6676 phpm 6688 snexxph 6766 sbthlem2 6774 niex 7021 enqex 7069 enq0ex 7148 npex 7182 ltnqex 7258 gtnqex 7259 recexprlemell 7331 recexprlemelu 7332 enrex 7433 axcnex 7546 peano5nnnn 7577 reex 7626 nnex 8584 zex 8915 qex 9274 ixxex 9523 serclim0 10913 climle 10942 iserabs 11083 isumshft 11098 explecnv 11113 prmex 11587 exmidunben 11731 istopon 11962 dmtopon 11972 lmres 12198 climcncf 12484 reldvg 12521 |
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