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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 4060 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴}) |
3 | abid1 2870 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
4 | 2, 3 | sseqtrrdi 4033 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 {cab 2709 ⊆ wss 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 |
This theorem is referenced by: dfopif 4870 opabssxpd 5723 fmpt 7111 eroprf 8811 cfslb2n 10265 rankcf 10774 genpv 10996 genpdm 10999 fimaxre3 12164 supadd 12186 supmul 12190 hashfacenOLD 14418 hashf1lem1OLD 14420 hashf1lem2 14421 mertenslem2 15835 4sqlem11 16892 lss1d 20718 lspsn 20757 lpval 22863 lpsscls 22865 ptuni2 23300 ptbasfi 23305 prdstopn 23352 xkopt 23379 tgpconncompeqg 23836 metrest 24253 mbfeqalem1 25382 limcfval 25613 nosupno 27430 nosupbday 27432 noinfno 27445 noinfbday 27447 addsproplem2 27680 addsuniflem 27711 negsid 27742 mulsproplem9 27807 ssltmul1 27829 ssltmul2 27830 precsexlem8 27887 precsexlem11 27890 nmosetre 30272 nmopsetretALT 31371 nmfnsetre 31385 sigaclcuni 33402 bnj849 34222 deranglem 34443 derangsn 34447 liness 35409 mblfinlem3 36830 ismblfin 36832 itg2addnclem 36842 areacirclem2 36880 sdclem2 36913 sdclem1 36914 ismtyval 36971 heibor1lem 36980 heibor1 36981 pmapglbx 38943 eldiophb 41797 hbtlem2 42168 oaun3lem1 42426 oaun3lem2 42427 upbdrech 44314 |
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