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Mirrors > Home > ILE Home > Th. List > ssexd | GIF version |
Description: A subclass of a set is a set. Deduction form of ssexg 4121. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssexd.1 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
ssexd.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
ssexd | ⊢ (𝜑 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexd.2 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | ssexd.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
3 | ssexg 4121 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2136 Vcvv 2726 ⊆ wss 3116 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 ax-sep 4100 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-in 3122 df-ss 3129 |
This theorem is referenced by: fex2 5356 riotaexg 5802 opabbrex 5886 funexw 6080 f1imaen2g 6759 fiss 6942 genipv 7450 suplocexprlemlub 7665 hashfacen 10749 ovshftex 10761 strslssd 12440 restid2 12565 2basgeng 12732 cnrest2 12886 cnptopresti 12888 cnptoprest 12889 cnptoprest2 12890 cnmpt2res 12947 psmetres2 12983 xmetres2 13029 limccnp2lem 13295 limccnp2cntop 13296 dvfvalap 13300 dvmulxxbr 13316 dvaddxx 13317 dvmulxx 13318 dviaddf 13319 dvimulf 13320 dvcoapbr 13321 dvmptaddx 13331 dvmptmulx 13332 |
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