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Mirrors > Home > MPE Home > Th. List > abssdv | Structured version Visualization version GIF version |
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) (Proof shortened by SN, 22-Dec-2024.) |
Ref | Expression |
---|---|
abssdv.1 | ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴)) |
Ref | Expression |
---|---|
abssdv | ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abssdv.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴)) | |
2 | 1 | ss2abdv 4061 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴}) |
3 | abid1 2871 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
4 | 2, 3 | sseqtrrdi 4034 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 {cab 2710 ⊆ wss 3949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 |
This theorem is referenced by: dfopif 4871 opabssxpd 5724 fmpt 7110 eroprf 8809 cfslb2n 10263 rankcf 10772 genpv 10994 genpdm 10997 fimaxre3 12160 supadd 12182 supmul 12186 hashfacenOLD 14414 hashf1lem1OLD 14416 hashf1lem2 14417 mertenslem2 15831 4sqlem11 16888 lss1d 20574 lspsn 20613 lpval 22643 lpsscls 22645 ptuni2 23080 ptbasfi 23085 prdstopn 23132 xkopt 23159 tgpconncompeqg 23616 metrest 24033 mbfeqalem1 25158 limcfval 25389 nosupno 27206 nosupbday 27208 noinfno 27221 noinfbday 27223 addsproplem2 27454 addsuniflem 27484 negsid 27515 mulsproplem9 27580 ssltmul1 27602 ssltmul2 27603 precsexlem8 27660 precsexlem11 27663 nmosetre 30017 nmopsetretALT 31116 nmfnsetre 31130 sigaclcuni 33116 bnj849 33936 deranglem 34157 derangsn 34161 liness 35117 mblfinlem3 36527 ismblfin 36529 itg2addnclem 36539 areacirclem2 36577 sdclem2 36610 sdclem1 36611 ismtyval 36668 heibor1lem 36677 heibor1 36678 pmapglbx 38640 eldiophb 41495 hbtlem2 41866 oaun3lem1 42124 oaun3lem2 42125 upbdrech 44015 |
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