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Mirrors > Home > MPE Home > Th. List > sseq2i | Structured version Visualization version GIF version |
Description: An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
sseq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
sseq2i | ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | sseq2 4022 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵) |
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: sseqtri 4032 sseqtrdi 4046 abss 4073 ssrab 4083 ssindif0 4470 difcom 4495 ssunsn2 4832 ssunpr 4839 sspr 4840 sstp 4841 ssintrab 4976 iunpwss 5112 propssopi 5518 dffun2OLDOLD 6575 ssimaex 6994 elpwun 7788 ssfi 9212 frfi 9319 alephislim 10121 cardaleph 10127 fin1a2lem12 10449 zornn0g 10543 ssxr 11328 nnwo 12953 isstruct 17186 issubmgm 18728 issubm 18829 grpissubg 19177 issubrng 20564 cntzsubrng 20584 islinds 21847 basdif0 22976 tgdif0 23015 cmpsublem 23423 cmpsub 23424 hauscmplem 23430 2ndcctbss 23479 fbncp 23863 cnextfval 24086 eltsms 24157 reconn 24864 cmssmscld 25398 nocvxminlem 27837 nocvxmin 27838 axcontlem3 28996 axcontlem4 28997 umgredg 29170 nbuhgr 29375 uhgrvd00 29567 vtxdginducedm1 29576 chsscon1i 31491 hatomistici 32391 chirredlem4 32422 atabs2i 32431 mdsymlem1 32432 mdsymlem3 32434 mdsymlem6 32437 mdsymlem8 32439 dmdbr5ati 32451 iundifdif 32583 poimir 37640 ismblfin 37648 cossssid2 38450 ntrk0kbimka 44029 ntrclsk3 44060 ntrneicls11 44080 wfaxrep 44950 wfaxsep 44951 abssf 45052 ssrabf 45054 stoweidlem57 46013 ovnsubadd 46528 ovnovollem3 46614 linccl 48260 lincdifsn 48270 |
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