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Mirrors > Home > ILE Home > Th. List > ssexg | GIF version |
Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
ssexg | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3091 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵)) | |
2 | 1 | imbi1d 230 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 → 𝐴 ∈ V) ↔ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V))) |
3 | vex 2663 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 3 | ssex 4035 | . . 3 ⊢ (𝐴 ⊆ 𝑥 → 𝐴 ∈ V) |
5 | 2, 4 | vtoclg 2720 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)) |
6 | 5 | impcom 124 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 |
This theorem is referenced by: ssexd 4038 difexg 4039 rabexg 4041 elssabg 4043 elpw2g 4051 abssexg 4076 snexg 4078 sess1 4229 sess2 4230 trsuc 4314 unexb 4333 abnexg 4337 uniexb 4364 xpexg 4623 riinint 4770 dmexg 4773 rnexg 4774 resexg 4829 resiexg 4834 imaexg 4863 exse2 4883 cnvexg 5046 coexg 5053 fabexg 5280 f1oabexg 5347 relrnfvex 5407 fvexg 5408 sefvex 5410 mptfvex 5474 mptexg 5613 ofres 5964 resfunexgALT 5976 cofunexg 5977 fnexALT 5979 f1dmex 5982 oprabexd 5993 mpoexxg 6076 tposexg 6123 frecabex 6263 erex 6421 mapex 6516 pmvalg 6521 elpmg 6526 elmapssres 6535 pmss12g 6537 ixpexgg 6584 ssdomg 6640 fiprc 6677 fival 6826 shftfvalg 10558 shftfval 10561 toponsspwpwg 12116 tgval 12145 tgvalex 12146 eltg 12148 eltg2 12149 tgss 12159 basgen2 12177 bastop1 12179 topnex 12182 resttopon 12267 restabs 12271 lmfval 12288 cnrest 12331 txss12 12362 metrest 12602 dvbss 12750 dvcnp2cntop 12759 dvaddxxbr 12761 dvmulxxbr 12762 |
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