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Theorem ce0nn 6180
Description: A natural raised to cardinal zero is nonempty. Theorem XI.2.44 of [Rosser] p. 383. (Contributed by SF, 9-Mar-2015.)
Assertion
Ref Expression
ce0nn (N Nn → (Nc 0c) ≠ )

Proof of Theorem ce0nn
Dummy variables t m n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . 6 m V
21elcompl 3225 . . . . 5 (m ∼ ((1st (2nd “ {0c})) “ ( FullFunc “ {})) ↔ ¬ m ((1st (2nd “ {0c})) “ ( FullFunc “ {})))
3 brres 4949 . . . . . . . . . 10 (t(1st (2nd “ {0c}))m ↔ (t1st m t (2nd “ {0c})))
4 eliniseg 5020 . . . . . . . . . . 11 (t (2nd “ {0c}) ↔ t2nd 0c)
54anbi2i 675 . . . . . . . . . 10 ((t1st m t (2nd “ {0c})) ↔ (t1st m t2nd 0c))
6 0cex 4392 . . . . . . . . . . 11 0c V
71, 6op1st2nd 5790 . . . . . . . . . 10 ((t1st m t2nd 0c) ↔ t = m, 0c)
83, 5, 73bitri 262 . . . . . . . . 9 (t(1st (2nd “ {0c}))mt = m, 0c)
98rexbii 2639 . . . . . . . 8 (t ( FullFunc “ {})t(1st (2nd “ {0c}))mt ( FullFunc “ {})t = m, 0c)
10 elima 4754 . . . . . . . 8 (m ((1st (2nd “ {0c})) “ ( FullFunc “ {})) ↔ t ( FullFunc “ {})t(1st (2nd “ {0c}))m)
11 risset 2661 . . . . . . . 8 (m, 0c ( FullFunc “ {}) ↔ t ( FullFunc “ {})t = m, 0c)
129, 10, 113bitr4i 268 . . . . . . 7 (m ((1st (2nd “ {0c})) “ ( FullFunc “ {})) ↔ m, 0c ( FullFunc “ {}))
13 eliniseg 5020 . . . . . . 7 (m, 0c ( FullFunc “ {}) ↔ m, 0c FullFunc )
141, 6brfullfunop 5867 . . . . . . 7 (m, 0c FullFunc ↔ (mc 0c) = )
1512, 13, 143bitri 262 . . . . . 6 (m ((1st (2nd “ {0c})) “ ( FullFunc “ {})) ↔ (mc 0c) = )
1615necon3bbii 2547 . . . . 5 m ((1st (2nd “ {0c})) “ ( FullFunc “ {})) ↔ (mc 0c) ≠ )
172, 16bitri 240 . . . 4 (m ∼ ((1st (2nd “ {0c})) “ ( FullFunc “ {})) ↔ (mc 0c) ≠ )
1817abbi2i 2464 . . 3 ∼ ((1st (2nd “ {0c})) “ ( FullFunc “ {})) = {m (mc 0c) ≠ }
19 1stex 4739 . . . . . 6 1st V
20 2ndex 5112 . . . . . . . 8 2nd V
2120cnvex 5102 . . . . . . 7 2nd V
22 snex 4111 . . . . . . 7 {0c} V
2321, 22imaex 4747 . . . . . 6 (2nd “ {0c}) V
2419, 23resex 5117 . . . . 5 (1st (2nd “ {0c})) V
25 ceex 6174 . . . . . . . 8 c V
2625fullfunex 5860 . . . . . . 7 FullFunc V
2726cnvex 5102 . . . . . 6 FullFunc V
28 snex 4111 . . . . . 6 {} V
2927, 28imaex 4747 . . . . 5 ( FullFunc “ {}) V
3024, 29imaex 4747 . . . 4 ((1st (2nd “ {0c})) “ ( FullFunc “ {})) V
3130complex 4104 . . 3 ∼ ((1st (2nd “ {0c})) “ ( FullFunc “ {})) V
3218, 31eqeltrri 2424 . 2 {m (mc 0c) ≠ } V
33 oveq1 5530 . . 3 (m = 0c → (mc 0c) = (0cc 0c))
3433neeq1d 2529 . 2 (m = 0c → ((mc 0c) ≠ ↔ (0cc 0c) ≠ ))
35 oveq1 5530 . . 3 (m = n → (mc 0c) = (nc 0c))
3635neeq1d 2529 . 2 (m = n → ((mc 0c) ≠ ↔ (nc 0c) ≠ ))
37 oveq1 5530 . . 3 (m = (n +c 1c) → (mc 0c) = ((n +c 1c) ↑c 0c))
3837neeq1d 2529 . 2 (m = (n +c 1c) → ((mc 0c) ≠ ↔ ((n +c 1c) ↑c 0c) ≠ ))
39 oveq1 5530 . . 3 (m = N → (mc 0c) = (Nc 0c))
4039neeq1d 2529 . 2 (m = N → ((mc 0c) ≠ ↔ (Nc 0c) ≠ ))
41 0cnc 6138 . . 3 0c NC
42 pw10 4161 . . . 4 1 =
43 nulel0c 4422 . . . 4 0c
4442, 43eqeltri 2423 . . 3 1 0c
45 ce0nnuli 6178 . . 3 ((0c NC 1 0c) → (0cc 0c) ≠ )
4641, 44, 45mp2an 653 . 2 (0cc 0c) ≠
47 nnnc 6146 . . 3 (n Nnn NC )
48 1cnc 6139 . . . . . 6 1c NC
49 0ex 4110 . . . . . . . 8 V
5049pw1sn 4165 . . . . . . 7 1{} = {{}}
5128snel1c 4140 . . . . . . 7 {{}} 1c
5250, 51eqeltri 2423 . . . . . 6 1{} 1c
53 ce0nnuli 6178 . . . . . 6 ((1c NC 1{} 1c) → (1cc 0c) ≠ )
5448, 52, 53mp2an 653 . . . . 5 (1cc 0c) ≠
5554jctr 526 . . . 4 ((nc 0c) ≠ → ((nc 0c) ≠ (1cc 0c) ≠ ))
56 ce0addcnnul 6179 . . . . 5 ((n NC 1c NC ) → (((n +c 1c) ↑c 0c) ≠ ↔ ((nc 0c) ≠ (1cc 0c) ≠ )))
5748, 56mpan2 652 . . . 4 (n NC → (((n +c 1c) ↑c 0c) ≠ ↔ ((nc 0c) ≠ (1cc 0c) ≠ )))
5855, 57syl5ibr 212 . . 3 (n NC → ((nc 0c) ≠ → ((n +c 1c) ↑c 0c) ≠ ))
5947, 58syl 15 . 2 (n Nn → ((nc 0c) ≠ → ((n +c 1c) ↑c 0c) ≠ ))
6032, 34, 36, 38, 40, 46, 59finds 4411 1 (N Nn → (Nc 0c) ≠ )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wa 358   = wceq 1642   wcel 1710  {cab 2339  wne 2516  wrex 2615  Vcvv 2859  ccompl 3205  c0 3550  {csn 3737  1cc1c 4134  1cpw1 4135   Nn cnnc 4373  0cc0c 4374   +c cplc 4375  cop 4561   class class class wbr 4639  1st c1st 4717  cima 4722  ccnv 4771   cres 4774  2nd c2nd 4783  (class class class)co 5525   FullFun cfullfun 5767   NC cncs 6088  c cce 6096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  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-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-uni 3892  df-int 3927  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-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-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-ce 6106
This theorem is referenced by:  ceclnn1  6189  nchoicelem5  6293
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