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 e. Nn /\ N e. Nn ) /\ ((M +o 1c) = (N +o 1c) /\ (M +o 1c) =/= (/))) -> M = N)

Proof of Theorem prepeano4

Dummy variables a b x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3559	. . 3 |- ((M +o 1c) =/= (/) <-> E.a a e. (M +o 1c))
2		elsuc 4413	. . . . 5 |- (a e. (M +o 1c) <-> E.b e. M E.x e. ∼ ba = (b u. {x}))
3		simplll 734	. . . . . . . 8 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> M e. Nn )
4		simpllr 735	. . . . . . . 8 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> N e. Nn )
5		simprl 732	. . . . . . . 8 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> b e. M)
6		simprr 733	. . . . . . . . . 10 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> x e. ∼ b)
7		vex 2862	. . . . . . . . . . 11 |- x e. _V
8	7	elcompl 3225	. . . . . . . . . 10 |- (x e. ∼ b <-> -. x e. b)
9	6, 8	sylib 188	. . . . . . . . 9 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> -. x e. b)
10	7	elsuci 4414	. . . . . . . . . . . 12 |- ((b e. M /\ -. x e. b) -> (b u. {x}) e. (M +o 1c))
11	8, 10	sylan2b 461	. . . . . . . . . . 11 |- ((b e. M /\ x e. ∼ b) -> (b u. {x}) e. (M +o 1c))
12	11	adantl 452	. . . . . . . . . 10 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> (b u. {x}) e. (M +o 1c))
13		simplr 731	. . . . . . . . . 10 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> (M +o 1c) = (N +o 1c))
14	12, 13	eleqtrd 2429	. . . . . . . . 9 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> (b u. {x}) e. (N +o 1c))
15		vex 2862	. . . . . . . . . 10 |- b e. _V
16	15, 7	nnsucelr 4428	. . . . . . . . 9 |- ((N e. Nn /\ (-. x e. b /\ (b u. {x}) e. (N +o 1c))) -> b e. N)
17	4, 9, 14, 16	syl12anc 1180	. . . . . . . 8 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> b e. N)
18		nnceleq 4430	. . . . . . . 8 |- (((M e. Nn /\ N e. Nn ) /\ (b e. M /\ b e. N)) -> M = N)
19	3, 4, 5, 17, 18	syl22anc 1183	. . . . . . 7 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> M = N)
20	19	a1d 22	. . . . . 6 |- ((((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) /\ (b e. M /\ x e. ∼ b)) -> (a = (b u. {x}) -> M = N))
21	20	rexlimdvva 2745	. . . . 5 |- (((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) -> (E.b e. M E.x e. ∼ ba = (b u. {x}) -> M = N))
22	2, 21	syl5bi 208	. . . 4 |- (((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) -> (a e. (M +o 1c) -> M = N))
23	22	exlimdv 1636	. . 3 |- (((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) -> (E.a a e. (M +o 1c) -> M = N))
24	1, 23	syl5bi 208	. 2 |- (((M e. Nn /\ N e. Nn ) /\ (M +o 1c) = (N +o 1c)) -> ((M +o 1c) =/= (/) -> M = N))
25	24	impr 602	1 |- (((M e. Nn /\ N e. Nn ) /\ ((M +o 1c) = (N +o 1c) /\ (M +o 1c) =/= (/))) -> M = N)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2516  E.wrex 2615   ∼ ccompl 3205   u. cun 3207  (/)c0 3550  {csn 3737  1_cc1c 4134   Nn cnnc 4373   +o cplc 4375

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-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