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Mirrors > Home > MPE Home > Th. List > difexi | Structured version Visualization version GIF version |
Description: Existence of a difference, inference version of difexg 5334. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.) |
Ref | Expression |
---|---|
difexi.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
difexi | ⊢ (𝐴 ∖ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexi.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | difexg 5334 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∖ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-in 3969 df-ss 3979 |
This theorem is referenced by: oev 8550 naddcllem 8712 sbthlem2 9122 findcard 9201 findcard2 9202 pssnn 9206 ssfi 9211 phplem2OLD 9252 phpOLD 9256 frfi 9318 unfilem3 9342 marypha1lem 9470 wemapso 9588 inf3lem3 9667 dfac9 10174 dfacacn 10179 kmlem11 10198 kmlem12 10199 fin23lem28 10377 isf32lem6 10395 isf32lem7 10396 isf32lem8 10397 domtriomlem 10479 axdc2lem 10485 axcclem 10494 zornn0g 10542 konigthlem 10605 grothprim 10871 hashbclem 14487 fi1uzind 14542 brfi1uzind 14543 brfi1indALT 14545 opfi1uzind 14546 ramub1lem1 17059 pltfval 18388 isirred 20435 cntzsdrg 20819 subdrgint 20820 lssset 20948 xrs1mnd 21439 xrs10 21440 xrs1cmn 21441 xrge0subm 21442 xrge0cmn 21443 cnmsgngrp 21614 psgninv 21617 psdmul 22187 neitr 23203 lecldbas 23242 imasdsf1olem 24398 xrge0gsumle 24868 xrge0tsms 24869 i1fd 25729 lhop1lem 26066 reefgim 26508 cxpcn2 26803 logbmpt 26845 newval 27908 newf 27911 addsval 28009 mulsval 28149 nnsex 28337 axlowdimlem15 28985 axlowdim 28990 elntg 29013 uhgrspan1lem1 29331 upgrres1lem1 29340 nbgrval 29367 nbfusgrlevtxm1 29408 cusgrfilem3 29489 vtxdginducedm1lem1 29571 vtxdginducedm1fi 29576 finsumvtxdg2ssteplem4 29580 rprmval 33523 dimkerim 33654 satfv1lem 35346 satfdm 35353 satffunlem1lem2 35387 satffunlem2lem2 35390 watvalN 39975 hvmapfval 41741 prjspval 42589 setindtr 43012 ssdifcl 43560 sssymdifcl 43561 clsk3nimkb 44029 iundjiunlem 46414 meaiuninclem 46435 meaiininclem 46441 lines 48580 |
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