Theorem slotstnscsi 12656

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 9001	. . . 4 |- 5 e. RR
2		5lt9 9122	. . . 4 |- 5 < 9
3	1, 2	gtneii 8056	. . 3 |- 9 =/= 5
4		tsetndx 12647	. . . 4 |- (TopSet ` ndx ) = 9
5		scandx 12612	. . . 4 |- (Scalar ` ndx ) = 5
6	4, 5	neeq12i 2364	. . 3 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) <-> 9 =/= 5 )
7	3, 6	mpbir 146	. 2 |- (TopSet ` ndx ) =/= (Scalar ` ndx )
8		6re 9003	. . . 4 |- 6 e. RR
9		6lt9 9121	. . . 4 |- 6 < 9
10	8, 9	gtneii 8056	. . 3 |- 9 =/= 6
11		vscandx 12618	. . . 4 |- ( .s ` ndx ) = 6
12	4, 11	neeq12i 2364	. . 3 |- ( (TopSet ` ndx ) =/= ( .s ` ndx ) <-> 9 =/= 6 )
13	10, 12	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .s ` ndx )
14		8re 9007	. . . 4 |- 8 e. RR
15		8lt9 9119	. . . 4 |- 8 < 9
16	14, 15	gtneii 8056	. . 3 |- 9 =/= 8
17		ipndx 12630	. . . 4 |- ( .i ` ndx ) = 8
18	4, 17	neeq12i 2364	. . 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 2347   ` cfv 5218   5c5 8976   6c6 8977   8c8 8979   9c9 8980   ndxcnx 12462  Scalarcsca 12542   .scvsca 12543   .icip 12544  TopSetcts 12545

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5881  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-7 8986  df-8 8987  df-9 8988  df-ndx 12468  df-slot 12469  df-sca 12555  df-vsca 12556  df-ip 12557  df-tset 12558

This theorem is referenced by:  sratsetg  13543