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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 4115. (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 4115 | . 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 2135 Vcvv 2721 ⊆ wss 3111 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 |
This theorem is referenced by: fex2 5350 riotaexg 5796 opabbrex 5877 f1imaen2g 6750 fiss 6933 genipv 7441 suplocexprlemlub 7656 hashfacen 10735 ovshftex 10747 strslssd 12383 restid2 12507 2basgeng 12629 cnrest2 12783 cnptopresti 12785 cnptoprest 12786 cnptoprest2 12787 cnmpt2res 12844 psmetres2 12880 xmetres2 12926 limccnp2lem 13192 limccnp2cntop 13193 dvfvalap 13197 dvmulxxbr 13213 dvaddxx 13214 dvmulxx 13215 dviaddf 13216 dvimulf 13217 dvcoapbr 13218 dvmptaddx 13228 dvmptmulx 13229 |
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