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Theorem ce0nn 6180
 Description: A natural raised to cardinal zero is non-empty. 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-1 5  ax-2 6  ax-3 7  ax-mp 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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