New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  prepeano4 GIF version

Theorem prepeano4 4451
 Description: Assuming a non-null successor, cardinal successor is one-to-one. Theorem X.1.19 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.)
Assertion
Ref Expression
prepeano4 (((M Nn N Nn ) ((M +c 1c) = (N +c 1c) (M +c 1c) ≠ )) → M = N)

Proof of Theorem prepeano4
Dummy variables a b x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3559 . . 3 ((M +c 1c) ≠ a a (M +c 1c))
2 elsuc 4413 . . . . 5 (a (M +c 1c) ↔ b M x ba = (b ∪ {x}))
3 simplll 734 . . . . . . . 8 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → M Nn )
4 simpllr 735 . . . . . . . 8 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → N Nn )
5 simprl 732 . . . . . . . 8 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → b M)
6 simprr 733 . . . . . . . . . 10 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → x b)
7 vex 2862 . . . . . . . . . . 11 x V
87elcompl 3225 . . . . . . . . . 10 (x b ↔ ¬ x b)
96, 8sylib 188 . . . . . . . . 9 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → ¬ x b)
107elsuci 4414 . . . . . . . . . . . 12 ((b M ¬ x b) → (b ∪ {x}) (M +c 1c))
118, 10sylan2b 461 . . . . . . . . . . 11 ((b M x b) → (b ∪ {x}) (M +c 1c))
1211adantl 452 . . . . . . . . . 10 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → (b ∪ {x}) (M +c 1c))
13 simplr 731 . . . . . . . . . 10 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → (M +c 1c) = (N +c 1c))
1412, 13eleqtrd 2429 . . . . . . . . 9 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → (b ∪ {x}) (N +c 1c))
15 vex 2862 . . . . . . . . . 10 b V
1615, 7nnsucelr 4428 . . . . . . . . 9 ((N Nn x b (b ∪ {x}) (N +c 1c))) → b N)
174, 9, 14, 16syl12anc 1180 . . . . . . . 8 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → b N)
18 nnceleq 4430 . . . . . . . 8 (((M Nn N Nn ) (b M b N)) → M = N)
193, 4, 5, 17, 18syl22anc 1183 . . . . . . 7 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → M = N)
2019a1d 22 . . . . . 6 ((((M Nn N Nn ) (M +c 1c) = (N +c 1c)) (b M x b)) → (a = (b ∪ {x}) → M = N))
2120rexlimdvva 2745 . . . . 5 (((M Nn N Nn ) (M +c 1c) = (N +c 1c)) → (b M x ba = (b ∪ {x}) → M = N))
222, 21syl5bi 208 . . . 4 (((M Nn N Nn ) (M +c 1c) = (N +c 1c)) → (a (M +c 1c) → M = N))
2322exlimdv 1636 . . 3 (((M Nn N Nn ) (M +c 1c) = (N +c 1c)) → (a a (M +c 1c) → M = N))
241, 23syl5bi 208 . 2 (((M Nn N Nn ) (M +c 1c) = (N +c 1c)) → ((M +c 1c) ≠ M = N))
2524impr 602 1 (((M Nn N Nn ) ((M +c 1c) = (N +c 1c) (M +c 1c) ≠ )) → M = N)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ∼ ccompl 3205   ∪ cun 3207  ∅c0 3550  {csn 3737  1cc1c 4134   Nn cnnc 4373   +c cplc 4375 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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-0c 4377  df-addc 4378  df-nnc 4379 This theorem is referenced by:  preaddccan2  4455  evenodddisj  4516  peano4  4557
 Copyright terms: Public domain W3C validator