Theorem suc11 3083

Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194.

Assertion

Ref	Expression
suc11	|- ((A e. On /\ B e. On) -> (suc A = suc B <-> A = B))

Proof of Theorem suc11

Step	Hyp	Ref	Expression
1		eloni 2948	. . . . 5 |- (A e. On -> Ord A)
2		ordn2lp 2958	. . . . . 6 |- (Ord A -> -. (A e. B /\ B e. A))
3		ianor 305	. . . . . 6 |- (-. (A e. B /\ B e. A) <-> (-. A e. B \/ -. B e. A))
4	2, 3	sylib 198	. . . . 5 |- (Ord A -> (-. A e. B \/ -. B e. A))
5	1, 4	syl 10	. . . 4 |- (A e. On -> (-. A e. B \/ -. B e. A))
6	5	adantr 389	. . 3 |- ((A e. On /\ B e. On) -> (-. A e. B \/ -. B e. A))
7		sucssel 3060	. . . . . 6 |- (A e. On -> (suc A (_ suc B -> A e. suc B))
8		eqimss 2099	. . . . . 6 |- (suc A = suc B -> suc A (_ suc B)
9	7, 8	syl5 21	. . . . 5 |- (A e. On -> (suc A = suc B -> A e. suc B))
10		elsuci 3025	. . . . . . 7 |- (A e. suc B -> (A e. B \/ A = B))
11	10	ord 232	. . . . . 6 |- (A e. suc B -> (-. A e. B -> A = B))
12	11	com12 11	. . . . 5 |- (-. A e. B -> (A e. suc B -> A = B))
13	9, 12	syl9 57	. . . 4 |- (A e. On -> (-. A e. B -> (suc A = suc B -> A = B)))
14		sucssel 3060	. . . . . 6 |- (B e. On -> (suc B (_ suc A -> B e. suc A))
15		eqimss2 2100	. . . . . 6 |- (suc A = suc B -> suc B (_ suc A)
16	14, 15	syl5 21	. . . . 5 |- (B e. On -> (suc A = suc B -> B e. suc A))
17		elsuci 3025	. . . . . . . 8 |- (B e. suc A -> (B e. A \/ B = A))
18	17	ord 232	. . . . . . 7 |- (B e. suc A -> (-. B e. A -> B = A))
19	18	com12 11	. . . . . 6 |- (-. B e. A -> (B e. suc A -> B = A))
20		eqcom 1469	. . . . . 6 |- (B = A <-> A = B)
21	19, 20	syl6ib 212	. . . . 5 |- (-. B e. A -> (B e. suc A -> A = B))
22	16, 21	syl9 57	. . . 4 |- (B e. On -> (-. B e. A -> (suc A = suc B -> A = B)))
23	13, 22	jaao 427	. . 3 |- ((A e. On /\ B e. On) -> ((-. A e. B \/ -. B e. A) -> (suc A = suc B -> A = B)))
24	6, 23	mpd 26	. 2 |- ((A e. On /\ B e. On) -> (suc A = suc B -> A = B))
25		suceq 3024	. 2 |- (A = B -> suc A = suc B)
26	24, 25	impbid1 515	1 |- ((A e. On /\ B e. On) -> (suc A = suc B <-> A = B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   (_ wss 2037  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  peano4 3142  limenpsi 4485

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

Copyright terms: Public domain