NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  nchoicelem19 GIF version

Theorem nchoicelem19 6307
Description: Lemma for nchoice 6308. Assuming well-ordering, there is a cardinal with a finite special set that is its own T-raising. Theorem 7.3 of [Specker] p. 974. (Contributed by SF, 20-Mar-2015.)
Assertion
Ref Expression
nchoicelem19 ( ≤c We NCm NC (( Spacm) Fin Tc m = m))

Proof of Theorem nchoicelem19
Dummy variables n x p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nchoicelem18 6306 . . 3 {x ( Spacx) Fin } V
2 fveq2 5328 . . . 4 (x = m → ( Spacx) = ( Spacm))
32eleq1d 2419 . . 3 (x = m → (( Spacx) Fin ↔ ( Spacm) Fin ))
4 fveq2 5328 . . . 4 (x = n → ( Spacx) = ( Spacn))
54eleq1d 2419 . . 3 (x = n → (( Spacx) Fin ↔ ( Spacn) Fin ))
6 id 19 . . 3 ( ≤c We NC → ≤c We NC )
7 vvex 4109 . . . . 5 V V
87ncelncsi 6121 . . . 4 Nc V NC
9 ltcpw1pwg 6202 . . . . . . . . 9 (V V → Nc 1V <c Nc V)
107, 9ax-mp 5 . . . . . . . 8 Nc 1V <c Nc V
11 df1c2 4168 . . . . . . . . 9 1c = 1V
1211nceqi 6109 . . . . . . . 8 Nc 1c = Nc 1V
13 pwv 3886 . . . . . . . . . 10 V = V
1413nceqi 6109 . . . . . . . . 9 Nc V = Nc V
1514eqcomi 2357 . . . . . . . 8 Nc V = Nc V
1610, 12, 153brtr4i 4667 . . . . . . 7 Nc 1c <c Nc V
17 nchoicelem8 6296 . . . . . . . 8 (( ≤c We NC Nc V NC ) → (¬ ( Nc V ↑c 0c) NCNc 1c <c Nc V))
188, 17mpan2 652 . . . . . . 7 ( ≤c We NC → (¬ ( Nc V ↑c 0c) NCNc 1c <c Nc V))
1916, 18mpbiri 224 . . . . . 6 ( ≤c We NC → ¬ ( Nc V ↑c 0c) NC )
20 nchoicelem3 6291 . . . . . 6 (( Nc V NC ¬ ( Nc V ↑c 0c) NC ) → ( SpacNc V) = { Nc V})
218, 19, 20sylancr 644 . . . . 5 ( ≤c We NC → ( SpacNc V) = { Nc V})
22 snfi 4431 . . . . 5 { Nc V} Fin
2321, 22syl6eqel 2441 . . . 4 ( ≤c We NC → ( SpacNc V) Fin )
24 fveq2 5328 . . . . . 6 (x = Nc V → ( Spacx) = ( SpacNc V))
2524eleq1d 2419 . . . . 5 (x = Nc V → (( Spacx) Fin ↔ ( SpacNc V) Fin ))
2625rspcev 2955 . . . 4 (( Nc V NC ( SpacNc V) Fin ) → x NC ( Spacx) Fin )
278, 23, 26sylancr 644 . . 3 ( ≤c We NCx NC ( Spacx) Fin )
281, 3, 5, 6, 27weds 5938 . 2 ( ≤c We NCm NC (( Spacm) Fin n NC (( Spacn) Finmc n)))
29 simpll 730 . . . . . . 7 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → ≤c We NC )
30 df-we 5906 . . . . . . . . . . 11 We = ( OrFr )
3130breqi 4645 . . . . . . . . . 10 ( ≤c We NC ↔ ≤c ( OrFr ) NC )
32 brin 4693 . . . . . . . . . 10 ( ≤c ( OrFr ) NC ↔ ( ≤c Or NC c Fr NC ))
3331, 32bitri 240 . . . . . . . . 9 ( ≤c We NC ↔ ( ≤c Or NC c Fr NC ))
3433simplbi 446 . . . . . . . 8 ( ≤c We NC → ≤c Or NC )
35 sopc 5934 . . . . . . . . . 10 ( ≤c Or NC ↔ ( ≤c Po NC c Connex NC ))
3635simplbi 446 . . . . . . . . 9 ( ≤c Or NC → ≤c Po NC )
37 porta 5933 . . . . . . . . . 10 ( ≤c Po NC ↔ ( ≤c Ref NC c Trans NC c Antisym NC ))
3837simp3bi 972 . . . . . . . . 9 ( ≤c Po NC → ≤c Antisym NC )
3936, 38syl 15 . . . . . . . 8 ( ≤c Or NC → ≤c Antisym NC )
4034, 39syl 15 . . . . . . 7 ( ≤c We NC → ≤c Antisym NC )
4129, 40syl 15 . . . . . 6 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → ≤c Antisym NC )
42 simplr 731 . . . . . . 7 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → m NC )
43 tccl 6160 . . . . . . 7 (m NCTc m NC )
4442, 43syl 15 . . . . . 6 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → Tc m NC )
45 simprr 733 . . . . . . . 8 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → n NC (( Spacn) Finmc n))
46 simprl 732 . . . . . . . . . 10 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → ( Spacm) Fin )
47 nchoicelem17 6305 . . . . . . . . . 10 (( ≤c We NC m NC ( Spacm) Fin ) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))
4829, 42, 46, 47syl3anc 1182 . . . . . . . . 9 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))
4948simpld 445 . . . . . . . 8 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → ( SpacTc m) Fin )
50 fveq2 5328 . . . . . . . . . . 11 (n = Tc m → ( Spacn) = ( SpacTc m))
5150eleq1d 2419 . . . . . . . . . 10 (n = Tc m → (( Spacn) Fin ↔ ( SpacTc m) Fin ))
52 breq2 4643 . . . . . . . . . 10 (n = Tc m → (mc nmc Tc m))
5351, 52imbi12d 311 . . . . . . . . 9 (n = Tc m → ((( Spacn) Finmc n) ↔ (( SpacTc m) Finmc Tc m)))
5453rspcv 2951 . . . . . . . 8 ( Tc m NC → (n NC (( Spacn) Finmc n) → (( SpacTc m) Finmc Tc m)))
5544, 45, 49, 54syl3c 57 . . . . . . 7 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → mc Tc m)
56 letc 6231 . . . . . . . . . 10 ((m NC m NC mc Tc m) → p NC m = Tc p)
57563expia 1153 . . . . . . . . 9 ((m NC m NC ) → (mc Tc mp NC m = Tc p))
5842, 42, 57syl2anc 642 . . . . . . . 8 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → (mc Tc mp NC m = Tc p))
59 nchoicelem12 6300 . . . . . . . . . . . . . . . 16 ((p NC ( SpacTc p) Fin ) → ( Spacp) Fin )
6059ad2ant2lr 728 . . . . . . . . . . . . . . 15 ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → ( Spacp) Fin )
61 fveq2 5328 . . . . . . . . . . . . . . . . . . . 20 (n = p → ( Spacn) = ( Spacp))
6261eleq1d 2419 . . . . . . . . . . . . . . . . . . 19 (n = p → (( Spacn) Fin ↔ ( Spacp) Fin ))
63 breq2 4643 . . . . . . . . . . . . . . . . . . 19 (n = p → ( Tc pc nTc pc p))
6462, 63imbi12d 311 . . . . . . . . . . . . . . . . . 18 (n = p → ((( Spacn) FinTc pc n) ↔ (( Spacp) FinTc pc p)))
6564rspcv 2951 . . . . . . . . . . . . . . . . 17 (p NC → (n NC (( Spacn) FinTc pc n) → (( Spacp) FinTc pc p)))
6665imp 418 . . . . . . . . . . . . . . . 16 ((p NC n NC (( Spacn) FinTc pc n)) → (( Spacp) FinTc pc p))
6766ad2ant2l 726 . . . . . . . . . . . . . . 15 ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → (( Spacp) FinTc pc p))
6860, 67mpd 14 . . . . . . . . . . . . . 14 ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → Tc pc p)
69 simplr 731 . . . . . . . . . . . . . . . 16 ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → p NC )
70 tccl 6160 . . . . . . . . . . . . . . . 16 (p NCTc p NC )
7169, 70syl 15 . . . . . . . . . . . . . . 15 ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → Tc p NC )
72 tlecg 6230 . . . . . . . . . . . . . . 15 (( Tc p NC p NC ) → ( Tc pc pTc Tc pc Tc p))
7371, 69, 72syl2anc 642 . . . . . . . . . . . . . 14 ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → ( Tc pc pTc Tc pc Tc p))
7468, 73mpbid 201 . . . . . . . . . . . . 13 ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → Tc Tc pc Tc p)
75 fveq2 5328 . . . . . . . . . . . . . . . . 17 (m = Tc p → ( Spacm) = ( SpacTc p))
7675eleq1d 2419 . . . . . . . . . . . . . . . 16 (m = Tc p → (( Spacm) Fin ↔ ( SpacTc p) Fin ))
77 breq1 4642 . . . . . . . . . . . . . . . . . 18 (m = Tc p → (mc nTc pc n))
7877imbi2d 307 . . . . . . . . . . . . . . . . 17 (m = Tc p → ((( Spacn) Finmc n) ↔ (( Spacn) FinTc pc n)))
7978ralbidv 2634 . . . . . . . . . . . . . . . 16 (m = Tc p → (n NC (( Spacn) Finmc n) ↔ n NC (( Spacn) FinTc pc n)))
8076, 79anbi12d 691 . . . . . . . . . . . . . . 15 (m = Tc p → ((( Spacm) Fin n NC (( Spacn) Finmc n)) ↔ (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))))
8180anbi2d 684 . . . . . . . . . . . . . 14 (m = Tc p → ((( ≤c We NC p NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) ↔ (( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n)))))
82 tceq 6158 . . . . . . . . . . . . . . 15 (m = Tc pTc m = Tc Tc p)
83 id 19 . . . . . . . . . . . . . . 15 (m = Tc pm = Tc p)
8482, 83breq12d 4652 . . . . . . . . . . . . . 14 (m = Tc p → ( Tc mc mTc Tc pc Tc p))
8581, 84imbi12d 311 . . . . . . . . . . . . 13 (m = Tc p → (((( ≤c We NC p NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → Tc mc m) ↔ ((( ≤c We NC p NC ) (( SpacTc p) Fin n NC (( Spacn) FinTc pc n))) → Tc Tc pc Tc p)))
8674, 85mpbiri 224 . . . . . . . . . . . 12 (m = Tc p → ((( ≤c We NC p NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → Tc mc m))
8786com12 27 . . . . . . . . . . 11 ((( ≤c We NC p NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → (m = Tc pTc mc m))
8887an32s 779 . . . . . . . . . 10 ((( ≤c We NC (( Spacm) Fin n NC (( Spacn) Finmc n))) p NC ) → (m = Tc pTc mc m))
8988rexlimdva 2738 . . . . . . . . 9 (( ≤c We NC (( Spacm) Fin n NC (( Spacn) Finmc n))) → (p NC m = Tc pTc mc m))
9089adantlr 695 . . . . . . . 8 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → (p NC m = Tc pTc mc m))
9158, 90syld 40 . . . . . . 7 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → (mc Tc mTc mc m))
9255, 91mpd 14 . . . . . 6 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → Tc mc m)
9341, 44, 42, 92, 55antid 5929 . . . . 5 ((( ≤c We NC m NC ) (( Spacm) Fin n NC (( Spacn) Finmc n))) → Tc m = m)
9493exp32 588 . . . 4 (( ≤c We NC m NC ) → (( Spacm) Fin → (n NC (( Spacn) Finmc n) → Tc m = m)))
9594imdistand 673 . . 3 (( ≤c We NC m NC ) → ((( Spacm) Fin n NC (( Spacn) Finmc n)) → (( Spacm) Fin Tc m = m)))
9695reximdva 2726 . 2 ( ≤c We NC → (m NC (( Spacm) Fin n NC (( Spacn) Finmc n)) → m NC (( Spacm) Fin Tc m = m)))
9728, 96mpd 14 1 ( ≤c We NCm NC (( Spacm) Fin Tc m = m))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wo 357   wa 358   = wceq 1642   wcel 1710  wral 2614  wrex 2615  Vcvv 2859  cin 3208  cpw 3722  {csn 3737  1cc1c 4134  1cpw1 4135  0cc0c 4374   +c cplc 4375   Fin cfin 4376   class class class wbr 4639  cfv 4781  (class class class)co 5525   Trans ctrans 5888   Ref cref 5889   Antisym cantisym 5890   Po cpartial 5891   Connex cconnex 5892   Or cstrict 5893   Fr cfound 5894   We cwe 5895   NC cncs 6088  c clec 6089   <c cltc 6090   Nc cnc 6091   Tc ctc 6093  2cc2c 6094  c cce 6096   Spac cspac 6273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-meredith 1406  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-tp 3743  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-ref 5900  df-antisym 5901  df-partial 5902  df-connex 5903  df-strict 5904  df-found 5905  df-we 5906  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-lec 6099  df-ltc 6100  df-nc 6101  df-tc 6103  df-2c 6104  df-3c 6105  df-ce 6106  df-tcfn 6107  df-spac 6274
This theorem is referenced by:  nchoice  6308
  Copyright terms: Public domain W3C validator