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Mirrors > Home > MPE Home > Th. List > ss2rabdv | Structured version Visualization version GIF version |
Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.) |
Ref | Expression |
---|---|
ss2rabdv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ss2rabdv | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2rabdv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) | |
2 | 1 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
3 | ss2rab 3641 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) | |
4 | 2, 3 | sylibr 223 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-in 3547 df-ss 3554 |
This theorem is referenced by: sess1 5006 suppfnss 7207 suppssov1 7214 suppssfv 7218 harword 8353 mrcss 16099 ablfac1b 18292 mptscmfsupp0 18751 lspss 18805 aspss 19153 dsmmacl 19904 dsmmsubg 19906 dsmmlss 19907 scmatdmat 20140 clsss 20668 lfinpfin 21137 qustgpopn 21733 metss2lem 22126 equivcau 22906 rrxmvallem 22995 ovolsslem 23059 itg2monolem1 23323 lgamucov 24564 sqff1o 24708 musum 24717 cusgrafilem1 26007 locfinreflem 29235 omsmon 29687 orvclteinc 29864 fin2solem 32565 poimirlem26 32605 poimirlem27 32606 cnambfre 32628 pclssN 34198 2polssN 34219 dihglblem3N 35602 dochss 35672 mapdordlem2 35944 nzss 37538 cusgrfilem1 40671 rmsuppss 41945 mndpsuppss 41946 scmsuppss 41947 |
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