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Mirrors > Home > MPE Home > Th. List > sseq1i | Structured version Visualization version GIF version |
Description: An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
sseq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
sseq1i | ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | sseq1 4021 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ss 3980 |
This theorem is referenced by: eqsstri 4030 eqsstrid 4044 ssab 4074 rabss 4082 uniiunlem 4097 prssg 4824 sstp 4841 tpss 4842 iunssf 5049 iunss 5050 pwtr 5463 iunopeqop 5531 pwssun 5580 cores2 6281 resssxp 6292 sspred 6332 dffun2OLDOLD 6575 sbcfg 6735 idref 7166 ovmptss 8117 fnsuppres 8215 frrlem7 8316 ordgt0ge1 8530 omopthlem1 8696 naddasslem1 8731 naddasslem2 8732 dmttrcl 9759 trcl 9766 rankbnd 9906 rankbnd2 9907 rankc1 9908 dfac12a 10187 fin23lem34 10384 axdc2lem 10486 alephval2 10610 indpi 10945 fsuppmapnn0fiublem 14028 prodeq1i 15949 0ram 17054 mreacs 17703 lsslinds 21869 2ndcctbss 23479 xkoinjcn 23711 restmetu 24599 xrlimcnp 27026 mptelee 28925 ausgrusgrb 29197 nbuhgr2vtx1edgblem 29383 nbgrsym 29395 isuvtx 29427 2wlkdlem6 29961 frcond1 30295 n4cyclfrgr 30320 shlesb1i 31415 mdsldmd1i 32360 csmdsymi 32363 lfuhgr 35102 2cycl2d 35124 dfon2lem3 35767 dfon2lem7 35771 cbvprodvw2 36230 filnetlem4 36364 ptrecube 37607 poimirlem30 37637 idinxpssinxp2 38300 cossssid2 38450 symrefref2 38545 redundeq1 38611 funALTVfun 38680 disjxrn 38728 omabs2 43322 undmrnresiss 43594 clcnvlem 43613 cnvtrrel 43660 brtrclfv2 43717 dfhe3 43765 dffrege76 43929 mnurndlem1 44277 ssabf 45040 rabssf 45059 imassmpt 45208 clnbgrsym 47762 sclnbgrelself 47772 setrec2 48926 |
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