Theorem peano4 4558

Description: The successor operation is one-to-one over the finite cardinals. Theorem X.1.66 of [Rosser] p. 537. (Contributed by SF, 20-Jan-2015.)

Assertion

Ref	Expression
peano4	|- ((M e. Nn /\ N e. Nn /\ (M +o 1c) = (N +o 1c)) -> M = N)

Proof of Theorem peano4

Step	Hyp	Ref	Expression
1		3simpa 952	. 2 |- ((M e. Nn /\ N e. Nn /\ (M +o 1c) = (N +o 1c)) -> (M e. Nn /\ N e. Nn ))
2		simp3 957	. 2 |- ((M e. Nn /\ N e. Nn /\ (M +o 1c) = (N +o 1c)) -> (M +o 1c) = (N +o 1c))
3		peano2 4404	. . . 4 |- (M e. Nn -> (M +o 1c) e. Nn )
4		nulnnn 4557	. . . . . 6 |- -. (/) e. Nn
5		eleq1 2413	. . . . . 6 |- ((M +o 1c) = (/) -> ((M +o 1c) e. Nn <-> (/) e. Nn ))
6	4, 5	mtbiri 294	. . . . 5 |- ((M +o 1c) = (/) -> -. (M +o 1c) e. Nn )
7	6	necon2ai 2562	. . . 4 |- ((M +o 1c) e. Nn -> (M +o 1c) =/= (/))
8	3, 7	syl 15	. . 3 |- (M e. Nn -> (M +o 1c) =/= (/))
9	8	3ad2ant1 976	. 2 |- ((M e. Nn /\ N e. Nn /\ (M +o 1c) = (N +o 1c)) -> (M +o 1c) =/= (/))
10		prepeano4 4452	. 2 |- (((M e. Nn /\ N e. Nn ) /\ ((M +o 1c) = (N +o 1c) /\ (M +o 1c) =/= (/))) -> M = N)
11	1, 2, 9, 10	syl12anc 1180	1 |- ((M e. Nn /\ N e. Nn /\ (M +o 1c) = (N +o 1c)) -> M = N)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  (/)c0 3551  1_cc1c 4135   Nn cnnc 4374   +o cplc 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448

This theorem is referenced by:  suc11nnc  4559  phi11lem1  4596  fnfreclem3  6320