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Mirrors > Home > ILE Home > Th. List > slotstnscsi | GIF version |
Description: The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
slotstnscsi | ⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5re 8996 | . . . 4 ⊢ 5 ∈ ℝ | |
2 | 5lt9 9117 | . . . 4 ⊢ 5 < 9 | |
3 | 1, 2 | gtneii 8051 | . . 3 ⊢ 9 ≠ 5 |
4 | tsetndx 12635 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
5 | scandx 12603 | . . . 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 8998 | . . . 4 ⊢ 6 ∈ ℝ | |
9 | 6lt9 9116 | . . . 4 ⊢ 6 < 9 | |
10 | 8, 9 | gtneii 8051 | . . 3 ⊢ 9 ≠ 6 |
11 | vscandx 12609 | . . . 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 9002 | . . . 4 ⊢ 8 ∈ ℝ | |
15 | 8lt9 9114 | . . . 4 ⊢ 8 < 9 | |
16 | 14, 15 | gtneii 8051 | . . 3 ⊢ 9 ≠ 8 |
17 | ipndx 12621 | . . . 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)) |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 978 ≠ wne 2347 ‘cfv 5216 5c5 8971 6c6 8972 8c8 8974 9c9 8975 ndxcnx 12453 Scalarcsca 12533 ·𝑠 cvsca 12534 ·𝑖cip 12535 TopSetcts 12536 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-7 8981 df-8 8982 df-9 8983 df-ndx 12459 df-slot 12460 df-sca 12546 df-vsca 12547 df-ip 12548 df-tset 12549 |
This theorem is referenced by: (None) |
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