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Theorem ssnn0ssfz 42127
Description: For any finite subset of NN0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 29549. (Contributed by AV, 30-Sep-2019.)
Assertion
Ref Expression
ssnn0ssfz |- ( A e. ( ~P NN0 i^i Fin ) -> E. n e. NN0 A C_ ( 0 ... n ) )
Distinct variable group:   A, n
Proof of Theorem ssnn0ssfz
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nn0 11307 . . 3 |- 0 e. NN0
2 simpr 477 . . . 4 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A = (/) ) -> A = (/) )
3 0ss 3972 . . . 4 |- (/) C_ ( 0 ... 0 )
42, 3syl6eqss 3655 . . 3 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A = (/) ) -> A C_ ( 0 ... 0 ) )
5 oveq2 6658 . . . . 5 |- ( n = 0 -> ( 0 ... n ) = ( 0 ... 0 ) )
65sseq2d 3633 . . . 4 |- ( n = 0 -> ( A C_ ( 0 ... n ) <-> A C_ ( 0 ... 0 ) ) )
76rspcev 3309 . . 3 |- ( ( 0 e. NN0 /\ A C_ ( 0 ... 0 ) ) -> E. n e. NN0 A C_ ( 0 ... n ) )
81, 4, 7sylancr 695 . 2 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A = (/) ) -> E. n e. NN0 A C_ ( 0 ... n ) )
9 elin 3796 . . . . . . 7 |- ( A e. ( ~P NN0 i^i Fin ) <-> ( A e. ~P NN0 /\ A e. Fin ) )
109simplbi 476 . . . . . 6 |- ( A e. ( ~P NN0 i^i Fin ) -> A e. ~P NN0 )
1110adantr 481 . . . . 5 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> A e. ~P NN0 )
1211elpwid 4170 . . . 4 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> A C_ NN0 )
13 nn0ssre 11296 . . . . . . 7 |- NN0 C_ RR
14 ltso 10118 . . . . . . 7 |- < Or RR
15 soss 5053 . . . . . . 7 |- ( NN0 C_ RR -> ( < Or RR -> < Or NN0 ) )
1613, 14, 15mp2 9 . . . . . 6 |- < Or NN0
1716a1i 11 . . . . 5 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> < Or NN0 )
189simprbi 480 . . . . . 6 |- ( A e. ( ~P NN0 i^i Fin ) -> A e. Fin )
1918adantr 481 . . . . 5 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> A e. Fin )
20 simpr 477 . . . . 5 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> A =/= (/) )
21 fisupcl 8375 . . . . 5 |- ( ( < Or NN0 /\ ( A e. Fin /\ A =/= (/) /\ A C_ NN0 ) ) -> sup ( A , NN0 , < ) e. A )
2217, 19, 20, 12, 21syl13anc 1328 . . . 4 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> sup ( A , NN0 , < ) e. A )
2312, 22sseldd 3604 . . 3 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> sup ( A , NN0 , < ) e. NN0 )
2412sselda 3603 . . . . . . 7 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> x e. NN0 )
25 nn0uz 11722 . . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
2624, 25syl6eleq 2711 . . . . . 6 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> x e. ( ZZ>= ` 0 ) )
2724nn0zd 11480 . . . . . . 7 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> x e. ZZ )
2812adantr 481 . . . . . . . . 9 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> A C_ NN0 )
2922adantr 481 . . . . . . . . 9 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> sup ( A , NN0 , < ) e. A )
3028, 29sseldd 3604 . . . . . . . 8 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> sup ( A , NN0 , < ) e. NN0 )
3130nn0zd 11480 . . . . . . 7 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> sup ( A , NN0 , < ) e. ZZ )
32 fisup2g 8374 . . . . . . . . . . . 12 |- ( ( < Or NN0 /\ ( A e. Fin /\ A =/= (/) /\ A C_ NN0 ) ) -> E. x e. A ( A. y e. A -. x < y /\ A. y e. NN0 ( y < x -> E. z e. A y < z ) ) )
3317, 19, 20, 12, 32syl13anc 1328 . . . . . . . . . . 11 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> E. x e. A ( A. y e. A -. x < y /\ A. y e. NN0 ( y < x -> E. z e. A y < z ) ) )
34 ssrexv 3667 . . . . . . . . . . 11 |- ( A C_ NN0 -> ( E. x e. A ( A. y e. A -. x < y /\ A. y e. NN0 ( y < x -> E. z e. A y < z ) ) -> E. x e. NN0 ( A. y e. A -. x < y /\ A. y e. NN0 ( y < x -> E. z e. A y < z ) ) ) )
3512, 33, 34sylc 65 . . . . . . . . . 10 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> E. x e. NN0 ( A. y e. A -. x < y /\ A. y e. NN0 ( y < x -> E. z e. A y < z ) ) )
3617, 35supub 8365 . . . . . . . . 9 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> ( x e. A -> -. sup ( A , NN0 , < ) < x ) )
3736imp 445 . . . . . . . 8 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> -. sup ( A , NN0 , < ) < x )
3824nn0red 11352 . . . . . . . . 9 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> x e. RR )
3930nn0red 11352 . . . . . . . . 9 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> sup ( A , NN0 , < ) e. RR )
4038, 39lenltd 10183 . . . . . . . 8 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> ( x <_ sup ( A , NN0 , < ) <-> -. sup ( A , NN0 , < ) < x ) )
4137, 40mpbird 247 . . . . . . 7 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> x <_ sup ( A , NN0 , < ) )
42 eluz2 11693 . . . . . . 7 |- ( sup ( A , NN0 , < ) e. ( ZZ>= ` x ) <-> ( x e. ZZ /\ sup ( A , NN0 , < ) e. ZZ /\ x <_ sup ( A , NN0 , < ) ) )
4327, 31, 41, 42syl3anbrc 1246 . . . . . 6 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> sup ( A , NN0 , < ) e. ( ZZ>= ` x ) )
44 eluzfz 12337 . . . . . 6 |- ( ( x e. ( ZZ>= ` 0 ) /\ sup ( A , NN0 , < ) e. ( ZZ>= ` x ) ) -> x e. ( 0 ... sup ( A , NN0 , < ) ) )
4526, 43, 44syl2anc 693 . . . . 5 |- ( ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) /\ x e. A ) -> x e. ( 0 ... sup ( A , NN0 , < ) ) )
4645ex 450 . . . 4 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> ( x e. A -> x e. ( 0 ... sup ( A , NN0 , < ) ) ) )
4746ssrdv 3609 . . 3 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> A C_ ( 0 ... sup ( A , NN0 , < ) ) )
48 oveq2 6658 . . . . 5 |- ( n = sup ( A , NN0 , < ) -> ( 0 ... n ) = ( 0 ... sup ( A , NN0 , < ) ) )
4948sseq2d 3633 . . . 4 |- ( n = sup ( A , NN0 , < ) -> ( A C_ ( 0 ... n ) <-> A C_ ( 0 ... sup ( A , NN0 , < ) ) ) )
5049rspcev 3309 . . 3 |- ( ( sup ( A , NN0 , < ) e. NN0 /\ A C_ ( 0 ... sup ( A , NN0 , < ) ) ) -> E. n e. NN0 A C_ ( 0 ... n ) )
5123, 47, 50syl2anc 693 . 2 |- ( ( A e. ( ~P NN0 i^i Fin ) /\ A =/= (/) ) -> E. n e. NN0 A C_ ( 0 ... n ) )
528, 51pm2.61dane 2881 1 |- ( A e. ( ~P NN0 i^i Fin ) -> E. n e. NN0 A C_ ( 0 ... n ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   Or wor 5034   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327
This theorem is referenced by: (None)
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