Theorem slotstnscsi 13228

Description: The slots Scalar, .s and .i are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)

Assertion

Ref	Expression
slotstnscsi	|- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )

Proof of Theorem slotstnscsi

Step	Hyp	Ref	Expression
1		5re 9189	. . . 4 |- 5 e. RR
2		5lt9 9311	. . . 4 |- 5 < 9
3	1, 2	gtneii 8242	. . 3 |- 9 =/= 5
4		tsetndx 13219	. . . 4 |- (TopSet ` ndx ) = 9
5		scandx 13184	. . . 4 |- (Scalar ` ndx ) = 5
6	4, 5	neeq12i 2417	. . 3 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) <-> 9 =/= 5 )
7	3, 6	mpbir 146	. 2 |- (TopSet ` ndx ) =/= (Scalar ` ndx )
8		6re 9191	. . . 4 |- 6 e. RR
9		6lt9 9310	. . . 4 |- 6 < 9
10	8, 9	gtneii 8242	. . 3 |- 9 =/= 6
11		vscandx 13190	. . . 4 |- ( .s ` ndx ) = 6
12	4, 11	neeq12i 2417	. . 3 |- ( (TopSet ` ndx ) =/= ( .s ` ndx ) <-> 9 =/= 6 )
13	10, 12	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .s ` ndx )
14		8re 9195	. . . 4 |- 8 e. RR
15		8lt9 9308	. . . 4 |- 8 < 9
16	14, 15	gtneii 8242	. . 3 |- 9 =/= 8
17		ipndx 13202	. . . 4 |- ( .i ` ndx ) = 8
18	4, 17	neeq12i 2417	. . 3 |- ( (TopSet ` ndx ) =/= ( .i ` ndx ) <-> 9 =/= 8 )
19	16, 18	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .i ` ndx )
20	7, 13, 19	3pm3.2i 1199	1 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )

Syntax hints:   /\ w3a 1002   =/= wne 2400   ` cfv 5318   5c5 9164   6c6 9165   8c8 9167   9c9 9168   ndxcnx 13029  Scalarcsca 13113   .scvsca 13114   .icip 13115  TopSetcts 13116

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-ndx 13035  df-slot 13036  df-sca 13126  df-vsca 13127  df-ip 13128  df-tset 13129

This theorem is referenced by:  sratsetg  14409