![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rabssdv | Structured version Visualization version GIF version |
Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.) |
Ref | Expression |
---|---|
rabssdv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
rabssdv | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabssdv.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵) | |
2 | 1 | 3exp 1118 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 ∈ 𝐵))) |
3 | 2 | ralrimiv 3143 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵)) |
4 | rabss 4082 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵)) | |
5 | 3, 4 | sylibr 234 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rab 3434 df-ss 3980 |
This theorem is referenced by: suppss2 8224 oemapvali 9722 cantnflem1 9727 harval2 10035 zsupss 12977 ramub1lem1 17060 symggen 19503 efgsfo 19772 ablfacrp 20101 ablfac1eu 20108 pgpfac1lem5 20114 ablfaclem3 20122 nrmr0reg 23773 ptcmplem3 24078 abelthlem2 26491 lgamgulmlem1 27087 sltonold 28298 rspectopn 33828 neibastop2lem 36343 topmeet 36347 weiunse 36451 cntotbnd 37783 mapdrvallem2 41628 aks6d1c6lem3 42154 onintunirab 43216 nadd2rabex 43376 k0004ss1 44141 liminfvalxr 45739 |
Copyright terms: Public domain | W3C validator |