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Theorem uzennn 10894
Description: An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
Assertion
Ref Expression
uzennn |- ( M e. ZZ -> ( ZZ>= ` M ) ~~ NN )
Proof of Theorem uzennn
Dummy variables x y j k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-uz 9938 . . . . 5 |- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } )
2 zex 9664 . . . . . 6 |- ZZ e. _V
32mptex 5946 . . . . 5 |- ( j e. ZZ |-> { k e. ZZ | j <_ k } ) e. _V
41, 3eqeltri 2311 . . . 4 |- ZZ>= e. _V
5 fvexg 5717 . . . 4 |- ( ( ZZ>= e. _V /\ M e. ZZ ) -> ( ZZ>= ` M ) e. _V )
64, 5mpan 428 . . 3 |- ( M e. ZZ -> ( ZZ>= ` M ) e. _V )
7 nn0ex 9580 . . . 4 |- NN0 e. _V
87a1i 9 . . 3 |- ( M e. ZZ -> NN0 e. _V )
9 eluzelz 9947 . . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
109adantl 277 . . . . . 6 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> x e. ZZ )
11 simpl 109 . . . . . 6 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> M e. ZZ )
1210, 11zsubcld 9784 . . . . 5 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> ( x - M ) e. ZZ )
13 eluzle 9950 . . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> M <_ x )
1413adantl 277 . . . . . 6 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> M <_ x )
1510zred 9779 . . . . . . 7 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> x e. RR )
1611zred 9779 . . . . . . 7 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> M e. RR )
1715, 16subge0d 8870 . . . . . 6 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> ( 0 <_ ( x - M ) <-> M <_ x ) )
1814, 17mpbird 167 . . . . 5 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> 0 <_ ( x - M ) )
19 elnn0z 9668 . . . . 5 |- ( ( x - M ) e. NN0 <-> ( ( x - M ) e. ZZ /\ 0 <_ ( x - M ) ) )
2012, 18, 19sylanbrc 421 . . . 4 |- ( ( M e. ZZ /\ x e. ( ZZ>= ` M ) ) -> ( x - M ) e. NN0 )
2120ex 115 . . 3 |- ( M e. ZZ -> ( x e. ( ZZ>= ` M ) -> ( x - M ) e. NN0 ) )
22 simpl 109 . . . . 5 |- ( ( M e. ZZ /\ y e. NN0 ) -> M e. ZZ )
23 nn0z 9675 . . . . . . 7 |- ( y e. NN0 -> y e. ZZ )
2423adantl 277 . . . . . 6 |- ( ( M e. ZZ /\ y e. NN0 ) -> y e. ZZ )
2524, 22zaddcld 9783 . . . . 5 |- ( ( M e. ZZ /\ y e. NN0 ) -> ( y + M ) e. ZZ )
26 nn0ge0 9599 . . . . . . 7 |- ( y e. NN0 -> 0 <_ y )
2726adantl 277 . . . . . 6 |- ( ( M e. ZZ /\ y e. NN0 ) -> 0 <_ y )
2822zred 9779 . . . . . . 7 |- ( ( M e. ZZ /\ y e. NN0 ) -> M e. RR )
2924zred 9779 . . . . . . 7 |- ( ( M e. ZZ /\ y e. NN0 ) -> y e. RR )
3028, 29addge02d 8869 . . . . . 6 |- ( ( M e. ZZ /\ y e. NN0 ) -> ( 0 <_ y <-> M <_ ( y + M ) ) )
3127, 30mpbid 147 . . . . 5 |- ( ( M e. ZZ /\ y e. NN0 ) -> M <_ ( y + M ) )
32 eluz2 9943 . . . . 5 |- ( ( y + M ) e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ ( y + M ) e. ZZ /\ M <_ ( y + M ) ) )
3322, 25, 31, 32syl3anbrc 1212 . . . 4 |- ( ( M e. ZZ /\ y e. NN0 ) -> ( y + M ) e. ( ZZ>= ` M ) )
3433ex 115 . . 3 |- ( M e. ZZ -> ( y e. NN0 -> ( y + M ) e. ( ZZ>= ` M ) ) )
359ad2antrl 494 . . . . . . 7 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> x e. ZZ )
3635zcnd 9780 . . . . . 6 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> x e. CC )
37 simpl 109 . . . . . . 7 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> M e. ZZ )
3837zcnd 9780 . . . . . 6 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> M e. CC )
39 simprr 537 . . . . . . 7 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> y e. NN0 )
4039nn0cnd 9633 . . . . . 6 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> y e. CC )
4136, 38, 40subadd2d 8663 . . . . 5 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> ( ( x - M ) = y <-> ( y + M ) = x ) )
42 bicom 140 . . . . . 6 |- ( ( ( x - M ) = y <-> ( y + M ) = x ) <-> ( ( y + M ) = x <-> ( x - M ) = y ) )
43 eqcom 2240 . . . . . . 7 |- ( ( y + M ) = x <-> x = ( y + M ) )
44 eqcom 2240 . . . . . . 7 |- ( ( x - M ) = y <-> y = ( x - M ) )
4543, 44bibi12i 229 . . . . . 6 |- ( ( ( y + M ) = x <-> ( x - M ) = y ) <-> ( x = ( y + M ) <-> y = ( x - M ) ) )
4642, 45bitri 184 . . . . 5 |- ( ( ( x - M ) = y <-> ( y + M ) = x ) <-> ( x = ( y + M ) <-> y = ( x - M ) ) )
4741, 46sylib 122 . . . 4 |- ( ( M e. ZZ /\ ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) ) -> ( x = ( y + M ) <-> y = ( x - M ) ) )
4847ex 115 . . 3 |- ( M e. ZZ -> ( ( x e. ( ZZ>= ` M ) /\ y e. NN0 ) -> ( x = ( y + M ) <-> y = ( x - M ) ) ) )
496, 8, 21, 34, 48en3d 7059 . 2 |- ( M e. ZZ -> ( ZZ>= ` M ) ~~ NN0 )
50 nn0ennn 10891 . 2 |- NN0 ~~ NN
51 entr 7075 . 2 |- ( ( ( ZZ>= ` M ) ~~ NN0 /\ NN0 ~~ NN ) -> ( ZZ>= ` M ) ~~ NN )
5249, 50, 51sylancl 417 1 |- ( M e. ZZ -> ( ZZ>= ` M ) ~~ NN )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   class class class wbr 4131   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   ~~ cen 7024   0cc0 8185   + caddc 8188   <_ cle 8367   - cmin 8504   NNcn 9313   NN0cn0 9574   ZZcz 9655   ZZ>=cuz 9937
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-er 6811  df-en 7027  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938
This theorem is used by:  xnn0nnen  10895  exmidunben  13375
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