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Mirrors > Home > ILE Home > Th. List > 2strbasg | Unicode version |
Description: The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
Ref | Expression |
---|---|
2str.g | |
2str.e | Slot |
2str.l | |
2str.n |
Ref | Expression |
---|---|
2strbasg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseslid 12025 | . 2 Slot | |
2 | 2str.g | . . 3 | |
3 | basendxnn 12024 | . . . . . 6 | |
4 | 3 | a1i 9 | . . . . 5 |
5 | simpl 108 | . . . . 5 | |
6 | opexg 4150 | . . . . 5 | |
7 | 4, 5, 6 | syl2anc 408 | . . . 4 |
8 | 2str.e | . . . . . . . 8 Slot | |
9 | 2str.n | . . . . . . . 8 | |
10 | 8, 9 | ndxarg 11992 | . . . . . . 7 |
11 | 10, 9 | eqeltri 2212 | . . . . . 6 |
12 | 11 | a1i 9 | . . . . 5 |
13 | simpr 109 | . . . . 5 | |
14 | opexg 4150 | . . . . 5 | |
15 | 12, 13, 14 | syl2anc 408 | . . . 4 |
16 | prexg 4133 | . . . 4 | |
17 | 7, 15, 16 | syl2anc 408 | . . 3 |
18 | 2, 17 | eqeltrid 2226 | . 2 |
19 | 3 | nnrei 8736 | . . . . . 6 |
20 | 2str.l | . . . . . . 7 | |
21 | basendx 12023 | . . . . . . 7 | |
22 | 20, 21, 10 | 3brtr4i 3958 | . . . . . 6 |
23 | 19, 22 | ltneii 7867 | . . . . 5 |
24 | 23 | a1i 9 | . . . 4 |
25 | funprg 5173 | . . . 4 | |
26 | 4, 12, 5, 13, 24, 25 | syl221anc 1227 | . . 3 |
27 | 2 | funeqi 5144 | . . 3 |
28 | 26, 27 | sylibr 133 | . 2 |
29 | prid1g 3627 | . . . 4 | |
30 | 7, 29 | syl 14 | . . 3 |
31 | 30, 2 | eleqtrrdi 2233 | . 2 |
32 | 1, 18, 28, 31 | strslfvd 12010 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wne 2308 cvv 2686 cpr 3528 cop 3530 class class class wbr 3929 wfun 5117 cfv 5123 c1 7628 clt 7807 cn 8727 cnx 11966 Slot cslot 11968 cbs 11969 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1re 7721 ax-addrcl 7724 ax-pre-ltirr 7739 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-pnf 7809 df-mnf 7810 df-ltxr 7812 df-inn 8728 df-ndx 11972 df-slot 11973 df-base 11975 |
This theorem is referenced by: grpbaseg 12077 eltpsg 12217 |
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