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Theorem nulnnn 4556
 Description: The empty class is not a natural. Theorem X.1.65 of [Rosser] p. 536. (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
nulnnn ¬ Nn

Proof of Theorem nulnnn
Dummy variables m n x a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 complab 3524 . . . . . . 7 ∼ {n n = } = {n ¬ n = }
2 df-sn 3741 . . . . . . . 8 {} = {n n = }
32compleqi 3244 . . . . . . 7 ∼ {} = ∼ {n n = }
4 df-ne 2518 . . . . . . . 8 (n ↔ ¬ n = )
54abbii 2465 . . . . . . 7 {n n} = {n ¬ n = }
61, 3, 53eqtr4ri 2384 . . . . . 6 {n n} = ∼ {}
7 snex 4111 . . . . . . 7 {} V
87complex 4104 . . . . . 6 ∼ {} V
96, 8eqeltri 2423 . . . . 5 {n n} V
10 neeq1 2524 . . . . 5 (n = 0c → (n ↔ 0c))
11 neeq1 2524 . . . . 5 (n = m → (nm))
12 neeq1 2524 . . . . 5 (n = (m +c 1c) → (n ↔ (m +c 1c) ≠ ))
13 neeq1 2524 . . . . 5 (n = x → (nx))
14 nulel0c 4422 . . . . . 6 0c
15 ne0i 3556 . . . . . 6 ( 0c → 0c)
1614, 15ax-mp 5 . . . . 5 0c
17 n0 3559 . . . . . 6 (ma a m)
18 vinf 4555 . . . . . . . . . . . . . . 15 ¬ V Fin
19 elunii 3896 . . . . . . . . . . . . . . . . . 18 ((V m m Nn ) → V Nn )
2019ancoms 439 . . . . . . . . . . . . . . . . 17 ((m Nn V m) → V Nn )
21 df-fin 4380 . . . . . . . . . . . . . . . . 17 Fin = Nn
2220, 21syl6eleqr 2444 . . . . . . . . . . . . . . . 16 ((m Nn V m) → V Fin )
2322ex 423 . . . . . . . . . . . . . . 15 (m Nn → (V m → V Fin ))
2418, 23mtoi 169 . . . . . . . . . . . . . 14 (m Nn → ¬ V m)
25 eleq1 2413 . . . . . . . . . . . . . . 15 (a = V → (a m ↔ V m))
2625notbid 285 . . . . . . . . . . . . . 14 (a = V → (¬ a m ↔ ¬ V m))
2724, 26syl5ibrcom 213 . . . . . . . . . . . . 13 (m Nn → (a = V → ¬ a m))
2827necon2ad 2564 . . . . . . . . . . . 12 (m Nn → (a ma ≠ V))
2928imp 418 . . . . . . . . . . 11 ((m Nn a m) → a ≠ V)
30 compleqb 3543 . . . . . . . . . . . 12 (a = V ↔ ∼ a = ∼ V)
3130necon3bii 2548 . . . . . . . . . . 11 (a ≠ V ↔ ∼ a ≠ ∼ V)
3229, 31sylib 188 . . . . . . . . . 10 ((m Nn a m) → ∼ a ≠ ∼ V)
33 complV 4070 . . . . . . . . . . 11 ∼ V =
3433neeq2i 2527 . . . . . . . . . 10 ( ∼ a ≠ ∼ V ↔ ∼ a)
3532, 34sylib 188 . . . . . . . . 9 ((m Nn a m) → ∼ a)
36 n0 3559 . . . . . . . . . 10 ( ∼ ax x a)
37 vex 2862 . . . . . . . . . . . . . . 15 x V
3837elcompl 3225 . . . . . . . . . . . . . 14 (x a ↔ ¬ x a)
3937elsuci 4414 . . . . . . . . . . . . . . 15 ((a m ¬ x a) → (a ∪ {x}) (m +c 1c))
40 ne0i 3556 . . . . . . . . . . . . . . 15 ((a ∪ {x}) (m +c 1c) → (m +c 1c) ≠ )
4139, 40syl 15 . . . . . . . . . . . . . 14 ((a m ¬ x a) → (m +c 1c) ≠ )
4238, 41sylan2b 461 . . . . . . . . . . . . 13 ((a m x a) → (m +c 1c) ≠ )
4342ex 423 . . . . . . . . . . . 12 (a m → (x a → (m +c 1c) ≠ ))
4443adantl 452 . . . . . . . . . . 11 ((m Nn a m) → (x a → (m +c 1c) ≠ ))
4544exlimdv 1636 . . . . . . . . . 10 ((m Nn a m) → (x x a → (m +c 1c) ≠ ))
4636, 45syl5bi 208 . . . . . . . . 9 ((m Nn a m) → ( ∼ a → (m +c 1c) ≠ ))
4735, 46mpd 14 . . . . . . . 8 ((m Nn a m) → (m +c 1c) ≠ )
4847ex 423 . . . . . . 7 (m Nn → (a m → (m +c 1c) ≠ ))
4948exlimdv 1636 . . . . . 6 (m Nn → (a a m → (m +c 1c) ≠ ))
5017, 49syl5bi 208 . . . . 5 (m Nn → (m → (m +c 1c) ≠ ))
519, 10, 11, 12, 13, 16, 50finds 4411 . . . 4 (x Nnx)
5251neneqd 2532 . . 3 (x Nn → ¬ x = )
5352nrex 2716 . 2 ¬ x Nn x =
54 risset 2661 . 2 ( Nnx Nn x = )
5553, 54mtbir 290 1 ¬ Nn
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∪ cun 3207  ∅c0 3550  {csn 3737  ∪cuni 3891  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   Fin cfin 4376 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 This theorem is referenced by:  peano4  4557  addccan2  4559  nntccl  6170
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