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Mirrors > Home > ILE Home > Th. List > 2strop1g | Unicode version |
Description: The other slot of a constructed two-slot structure. Version of 2stropg 12263 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
Ref | Expression |
---|---|
2str1.g | |
2str1.b | |
2str1.n | |
2str1.e | Slot |
Ref | Expression |
---|---|
2strop1g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2str1.e | . . 3 Slot | |
2 | 2str1.n | . . 3 | |
3 | 1, 2 | ndxslid 12186 | . 2 Slot |
4 | 2str1.g | . . 3 | |
5 | 2str1.b | . . 3 | |
6 | 4, 5, 2 | 2strstr1g 12264 | . 2 Struct |
7 | simpr 109 | . 2 | |
8 | opexg 4188 | . . . . 5 | |
9 | 2, 7, 8 | sylancr 411 | . . . 4 |
10 | prid2g 3664 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | 1, 2 | ndxarg 12184 | . . . 4 |
13 | 12 | opeq1i 3744 | . . 3 |
14 | 11, 13, 4 | 3eltr4g 2243 | . 2 |
15 | 3, 6, 7, 14 | opelstrsl 12257 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cvv 2712 cpr 3561 cop 3563 class class class wbr 3965 cfv 5169 clt 7906 cn 8827 cnx 12158 Slot cslot 12160 cbs 12161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-addcom 7826 ax-addass 7828 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-0id 7834 ax-rnegex 7835 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-inn 8828 df-n0 9085 df-z 9162 df-uz 9434 df-fz 9906 df-struct 12163 df-ndx 12164 df-slot 12165 df-base 12167 |
This theorem is referenced by: (None) |
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