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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 3106 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
3 | ss2rab 3998 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) | |
4 | 2, 3 | sylibr 237 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3062 {crab 3066 ⊆ wss 3880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rab 3071 df-v 3422 df-in 3887 df-ss 3897 |
This theorem is referenced by: rabssrabd 4010 sess1 5533 suppssov1 7960 suppssfv 7964 harword 9203 mrcss 17143 ablfac1b 19481 mptscmfsupp0 19988 lspss 20045 dsmmacl 20727 dsmmsubg 20729 dsmmlss 20730 aspss 20860 scmatdmat 21436 clsss 21975 lfinpfin 22445 qustgpopn 23041 metss2lem 23433 equivcau 24221 rrxmvallem 24325 ovolsslem 24405 itg2monolem1 24672 lgamucov 25944 sqff1o 26088 musum 26097 cusgrfilem1 27567 clwlknf1oclwwlknlem3 28190 rmfsupp2 31235 locfinreflem 31528 omsmon 32001 orvclteinc 32178 naddssim 33600 madess 33822 cofcut1 33853 fin2solem 35526 poimirlem26 35566 poimirlem27 35567 cnambfre 35588 pclssN 37671 2polssN 37692 dihglblem3N 39072 dochss 39142 mapdordlem2 39414 nna4b4nsq 40228 nzss 41636 rmsuppss 45407 mndpsuppss 45408 scmsuppss 45409 |
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