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Mirrors > Home > MPE Home > Th. List > eqssi | Structured version Visualization version GIF version |
Description: Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.) |
Ref | Expression |
---|---|
eqssi.1 | ⊢ 𝐴 ⊆ 𝐵 |
eqssi.2 | ⊢ 𝐵 ⊆ 𝐴 |
Ref | Expression |
---|---|
eqssi | ⊢ 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqssi.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | eqssi.2 | . 2 ⊢ 𝐵 ⊆ 𝐴 | |
3 | eqss 3982 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 |
This theorem is referenced by: inv1 4348 unv 4349 intab 4906 intabs 5245 intid 5350 dmv 5792 0ima 5946 cnvrescnv 6052 find 7607 dftpos4 7911 wfrlem16 7970 dfom3 9110 tc2 9184 tcidm 9188 tc0 9189 rankuni 9292 rankval4 9296 djuunxp 9350 djuun 9355 ackbij1 9660 cfom 9686 fin23lem16 9757 itunitc 9843 inaprc 10258 nqerf 10352 dmrecnq 10390 dmaddsr 10507 dmmulsr 10508 axaddf 10567 axmulf 10568 dfnn2 11651 dfuzi 12074 unirnioo 12838 uzrdgfni 13327 0bits 15788 4sqlem19 16299 ledm 17834 lern 17835 efgsfo 18865 0frgp 18905 indiscld 21699 leordtval2 21820 lecldbas 21827 llyidm 22096 nllyidm 22097 toplly 22098 lly1stc 22104 txuni2 22173 txindis 22242 ust0 22828 qdensere 23378 xrtgioo 23414 zdis 23424 xrhmeo 23550 bndth 23562 ismbf3d 24255 dvef 24577 reeff1o 25035 efifo 25131 dvloglem 25231 logf1o2 25233 choc1 29104 shsidmi 29161 shsval2i 29164 omlsii 29180 chdmm1i 29254 chj1i 29266 chm0i 29267 shjshsi 29269 span0 29319 spanuni 29321 sshhococi 29323 spansni 29334 pjoml4i 29364 pjrni 29479 shatomistici 30138 sumdmdlem2 30196 rinvf1o 30375 sigapildsys 31421 sxbrsigalem0 31529 dya2iocucvr 31542 sxbrsigalem4 31545 sxbrsiga 31548 ballotth 31795 kur14lem6 32458 mrsubrn 32760 msubrn 32776 filnetlem3 33728 filnetlem4 33729 onint1 33797 oninhaus 33798 bj-rabtr 34251 bj-rabtrAUTO 34253 bj-disj2r 34343 bj-nuliotaALT 34354 bj-idres 34455 icoreunrn 34643 comptiunov2i 40071 unisnALT 41280 fsumiunss 41876 fourierdlem62 42473 fouriersw 42536 salexct 42637 salgencntex 42646 |
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