Theorem slotstnscsi 13408

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 9316	. . . 4 |- 5 e. RR
2		5lt9 9438	. . . 4 |- 5 < 9
3	1, 2	gtneii 8369	. . 3 |- 9 =/= 5
4		tsetndx 13399	. . . 4 |- (TopSet ` ndx ) = 9
5		scandx 13364	. . . 4 |- (Scalar ` ndx ) = 5
6	4, 5	neeq12i 2429	. . 3 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) <-> 9 =/= 5 )
7	3, 6	mpbir 146	. 2 |- (TopSet ` ndx ) =/= (Scalar ` ndx )
8		6re 9318	. . . 4 |- 6 e. RR
9		6lt9 9437	. . . 4 |- 6 < 9
10	8, 9	gtneii 8369	. . 3 |- 9 =/= 6
11		vscandx 13370	. . . 4 |- ( .s ` ndx ) = 6
12	4, 11	neeq12i 2429	. . 3 |- ( (TopSet ` ndx ) =/= ( .s ` ndx ) <-> 9 =/= 6 )
13	10, 12	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .s ` ndx )
14		8re 9322	. . . 4 |- 8 e. RR
15		8lt9 9435	. . . 4 |- 8 < 9
16	14, 15	gtneii 8369	. . 3 |- 9 =/= 8
17		ipndx 13382	. . . 4 |- ( .i ` ndx ) = 8
18	4, 17	neeq12i 2429	. . 3 |- ( (TopSet ` ndx ) =/= ( .i ` ndx ) <-> 9 =/= 8 )
19	16, 18	mpbir 146	. 2 |- (TopSet ` ndx ) =/= ( .i ` ndx )
20	7, 13, 19	3pm3.2i 1202	1 |- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )

Syntax hints:   /\ w3a 1005   =/= wne 2412   ` cfv 5352   5c5 9291   6c6 9292   8c8 9294   9c9 9295   ndxcnx 13209  Scalarcsca 13293   .scvsca 13294   .icip 13295  TopSetcts 13296

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-ndx 13215  df-slot 13216  df-sca 13306  df-vsca 13307  df-ip 13308  df-tset 13309

This theorem is referenced by:  sratsetg  14593