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Theorem fzss2 9029
Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
fzss2  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( M ... K )  C_  ( M ... N ) )

Proof of Theorem fzss2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elfzuz 8988 . . . . 5  |-  ( k  e.  ( M ... K )  ->  k  e.  ( ZZ>= `  M )
)
21adantl 266 . . . 4  |-  ( ( N  e.  ( ZZ>= `  K )  /\  k  e.  ( M ... K
) )  ->  k  e.  ( ZZ>= `  M )
)
3 elfzuz3 8989 . . . . 5  |-  ( k  e.  ( M ... K )  ->  K  e.  ( ZZ>= `  k )
)
4 uztrn 8585 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  k )
)  ->  N  e.  ( ZZ>= `  k )
)
53, 4sylan2 274 . . . 4  |-  ( ( N  e.  ( ZZ>= `  K )  /\  k  e.  ( M ... K
) )  ->  N  e.  ( ZZ>= `  k )
)
6 elfzuzb 8986 . . . 4  |-  ( k  e.  ( M ... N )  <->  ( k  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  k ) ) )
72, 5, 6sylanbrc 402 . . 3  |-  ( ( N  e.  ( ZZ>= `  K )  /\  k  e.  ( M ... K
) )  ->  k  e.  ( M ... N
) )
87ex 112 . 2  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( k  e.  ( M ... K
)  ->  k  e.  ( M ... N ) ) )
98ssrdv 2979 1  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( M ... K )  C_  ( M ... N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    e. wcel 1409    C_ wss 2945   ` cfv 4930  (class class class)co 5540   ZZ>=cuz 8569   ...cfz 8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltwlin 7055
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-neg 7248  df-z 8303  df-uz 8570  df-fz 8977
This theorem is referenced by:  fzssp1  9032  elfz0add  9081  elfz0addOLD  9082  fzoss2  9130  bcm1k  9628
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