Theorem slotstnscsi 12872

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 9069	. . . 4 |- 5 e. RR
2		5lt9 9191	. . . 4 |- 5 < 9
3	1, 2	gtneii 8122	. . 3 |- 9 =/= 5
4		tsetndx 12863	. . . 4 |- (TopSet ` ndx ) = 9
5		scandx 12828	. . . 4 |- (Scalar ` ndx ) = 5
6	4, 5	neeq12i 2384	. . 3 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) <-> 9 =/= 5 )
7	3, 6	mpbir 146	. 2 |- (TopSet ` ndx ) =/= (Scalar ` ndx )
8		6re 9071	. . . 4 |- 6 e. RR
9		6lt9 9190	. . . 4 |- 6 < 9
10	8, 9	gtneii 8122	. . 3 |- 9 =/= 6
11		vscandx 12834	. . . 4 |- ( .s ` ndx ) = 6
12	4, 11	neeq12i 2384	. . 3 |- ( (TopSet ` ndx ) =/= ( .s ` ndx ) <-> 9 =/= 6 )
13	10, 12	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .s ` ndx )
14		8re 9075	. . . 4 |- 8 e. RR
15		8lt9 9188	. . . 4 |- 8 < 9
16	14, 15	gtneii 8122	. . 3 |- 9 =/= 8
17		ipndx 12846	. . . 4 |- ( .i ` ndx ) = 8
18	4, 17	neeq12i 2384	. . 3 |- ( (TopSet ` ndx ) =/= ( .i ` ndx ) <-> 9 =/= 8 )
19	16, 18	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .i ` ndx )
20	7, 13, 19	3pm3.2i 1177	1 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )

Syntax hints:   /\ w3a 980   =/= wne 2367   ` cfv 5258   5c5 9044   6c6 9045   8c8 9047   9c9 9048   ndxcnx 12675  Scalarcsca 12758   .scvsca 12759   .icip 12760  TopSetcts 12761

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-ndx 12681  df-slot 12682  df-sca 12771  df-vsca 12772  df-ip 12773  df-tset 12774

This theorem is referenced by:  sratsetg  14001