Theorem slotstnscsi 12589

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 8969	. . . 4 |- 5 e. RR
2		5lt9 9090	. . . 4 |- 5 < 9
3	1, 2	gtneii 8027	. . 3 |- 9 =/= 5
4		tsetndx 12580	. . . 4 |- (TopSet ` ndx ) = 9
5		scandx 12556	. . . 4 |- (Scalar ` ndx ) = 5
6	4, 5	neeq12i 2362	. . 3 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) <-> 9 =/= 5 )
7	3, 6	mpbir 146	. 2 |- (TopSet ` ndx ) =/= (Scalar ` ndx )
8		6re 8971	. . . 4 |- 6 e. RR
9		6lt9 9089	. . . 4 |- 6 < 9
10	8, 9	gtneii 8027	. . 3 |- 9 =/= 6
11		vscandx 12562	. . . 4 |- ( .s ` ndx ) = 6
12	4, 11	neeq12i 2362	. . 3 |- ( (TopSet ` ndx ) =/= ( .s ` ndx ) <-> 9 =/= 6 )
13	10, 12	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .s ` ndx )
14		8re 8975	. . . 4 |- 8 e. RR
15		8lt9 9087	. . . 4 |- 8 < 9
16	14, 15	gtneii 8027	. . 3 |- 9 =/= 8
17		ipndx 12570	. . . 4 |- ( .i ` ndx ) = 8
18	4, 17	neeq12i 2362	. . 3 |- ( (TopSet ` ndx ) =/= ( .i ` ndx ) <-> 9 =/= 8 )
19	16, 18	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .i ` ndx )
20	7, 13, 19	3pm3.2i 1175	1 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )

Syntax hints:   /\ w3a 978   =/= wne 2345   ` cfv 5208   5c5 8944   6c6 8945   8c8 8947   9c9 8948   ndxcnx 12424  Scalarcsca 12494   .scvsca 12495   .icip 12496  TopSetcts 12497

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-lttrn 7900  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fv 5216  df-ov 5868  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-5 8952  df-6 8953  df-7 8954  df-8 8955  df-9 8956  df-ndx 12430  df-slot 12431  df-sca 12507  df-vsca 12508  df-ip 12509  df-tset 12510

This theorem is referenced by: (None)