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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 5328. (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 5328 | . 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 2107 Vcvv 3475 ∖ cdif 3946 |
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 ax-sep 5300 |
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-rab 3434 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 |
This theorem is referenced by: oev 8514 naddcllem 8675 sbthlem2 9084 findcard 9163 findcard2 9164 pssnn 9168 ssfi 9173 phplem2OLD 9218 phpOLD 9222 pssnnOLD 9265 findcard2OLD 9284 frfi 9288 unfilem3 9312 marypha1lem 9428 wemapso 9546 inf3lem3 9625 dfac9 10131 dfacacn 10136 kmlem11 10155 kmlem12 10156 fin23lem28 10335 isf32lem6 10353 isf32lem7 10354 isf32lem8 10355 domtriomlem 10437 axdc2lem 10443 axcclem 10452 zornn0g 10500 konigthlem 10563 grothprim 10829 hashbclem 14411 fi1uzind 14458 brfi1uzind 14459 brfi1indALT 14461 opfi1uzind 14462 ramub1lem1 16959 pltfval 18284 isirred 20233 cntzsdrg 20418 subdrgint 20419 lssset 20544 xrs1mnd 20983 xrs10 20984 xrs1cmn 20985 xrge0subm 20986 xrge0cmn 20987 cnmsgngrp 21132 psgninv 21135 neitr 22684 lecldbas 22723 imasdsf1olem 23879 xrge0gsumle 24349 xrge0tsms 24350 i1fd 25198 lhop1lem 25530 reefgim 25962 cxpcn2 26254 logbmpt 26293 newval 27350 newf 27353 addsval 27446 mulsval 27565 axlowdimlem15 28214 axlowdim 28219 elntg 28242 uhgrspan1lem1 28557 upgrres1lem1 28566 nbgrval 28593 nbfusgrlevtxm1 28634 cusgrfilem3 28714 vtxdginducedm1lem1 28796 vtxdginducedm1fi 28801 finsumvtxdg2ssteplem4 28805 rprmval 32633 dimkerim 32712 satfv1lem 34353 satfdm 34360 satffunlem1lem2 34394 satffunlem2lem2 34397 watvalN 38864 hvmapfval 40630 prjspval 41345 setindtr 41763 ssdifcl 42322 sssymdifcl 42323 clsk3nimkb 42791 iundjiunlem 45175 meaiuninclem 45196 meaiininclem 45202 lines 47417 |
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