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Mirrors > Home > ILE Home > Th. List > sseq1 | Unicode version |
Description: Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Ref | Expression |
---|---|
sseq1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 3171 |
. 2
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2 | sstr2 3163 |
. . . 4
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3 | 2 | adantl 277 |
. . 3
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4 | sstr2 3163 |
. . . 4
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5 | 4 | adantr 276 |
. . 3
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6 | 3, 5 | impbid 129 |
. 2
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7 | 1, 6 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3136 df-ss 3143 |
This theorem is referenced by: sseq12 3181 sseq1i 3182 sseq1d 3185 nssne2 3215 sbss 3532 pwjust 3577 elpw 3582 elpwg 3584 sssnr 3754 ssprr 3757 sstpr 3758 unimax 3844 trss 4111 elssabg 4149 bnd2 4174 exmidexmid 4197 exmidsssn 4203 exmidsssnc 4204 exmid1stab 4209 mss 4227 exss 4228 frforeq2 4346 ordtri2orexmid 4523 ontr2exmid 4525 onsucsssucexmid 4527 reg2exmidlema 4534 sucprcreg 4549 ordtri2or2exmid 4571 ontri2orexmidim 4572 onintexmid 4573 tfis 4583 tfisi 4587 elomssom 4605 nnregexmid 4621 releq 4709 xpsspw 4739 iss 4954 relcnvtr 5149 iotass 5196 fununi 5285 funcnvuni 5286 funimaexglem 5300 ffoss 5494 ssimaex 5578 tfrlem1 6309 el2oss1o 6444 nnsucsssuc 6493 qsss 6594 phpm 6865 ssfiexmid 6876 findcard2d 6891 findcard2sd 6892 diffifi 6894 isinfinf 6897 fiintim 6928 fisseneq 6931 fidcenumlemrk 6953 fidcenumlemr 6954 sbthlem2 6957 isbth 6966 ctssdclemr 7111 onntri45 7240 tapeq1 7251 elinp 7473 sup3exmid 8914 zfz1isolem1 10820 zfz1iso 10821 fimaxre2 11235 sumeq1 11363 fsum2d 11443 fsumabs 11473 fsumiun 11485 prodeq1f 11560 fprod2d 11631 exmidunben 12427 ctiunct 12441 ssomct 12446 restsspw 12698 uniopn 13504 fiinopn 13507 fiinbas 13552 baspartn 13553 eltg2 13556 eltg3 13560 topbas 13570 clsval 13614 neival 13646 neiint 13648 neipsm 13657 opnneissb 13658 opnssneib 13659 innei 13666 restbasg 13671 cnpdis 13745 txbas 13761 eltx 13762 neitx 13771 txlm 13782 blssexps 13932 blssex 13933 neibl 13994 metrest 14009 xmettx 14013 tgioo 14049 tgqioo 14050 limcimolemlt 14136 recnprss 14159 bj-om 14692 bj-2inf 14693 bj-nntrans 14706 bj-omtrans 14711 subctctexmid 14753 pw1nct 14755 |
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