Theorem suc11 2341

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 2209	. . . . 5 |- (A e. On -> Ord A)
2		ordn2lp 2219	. . . . . 6 |- (Ord A -> -. (A e. B /\ B e. A))
3		ianor 253	. . . . . 6 |- (-. (A e. B /\ B e. A) <-> (-. A e. B \/ -. B e. A))
4	2, 3	sylib 173	. . . . 5 |- (Ord A -> (-. A e. B \/ -. B e. A))
5	1, 4	syl 12	. . . 4 |- (A e. On -> (-. A e. B \/ -. B e. A))
6	5	adantr 306	. . 3 |- ((A e. On /\ B e. On) -> (-. A e. B \/ -. B e. A))
7		sucssel 2321	. . . . . 6 |- (A e. On -> (suc A (_ suc B -> A e. suc B))
8		eqimss 1548	. . . . . 6 |- (suc A = suc B -> suc A (_ suc B)
9	7, 8	syl5 22	. . . . 5 |- (A e. On -> (suc A = suc B -> A e. suc B))
10		elsuci 2289	. . . . . . 7 |- (A e. suc B -> (A e. B \/ A = B))
11	10	ord 202	. . . . . 6 |- (A e. suc B -> (-. A e. B -> A = B))
12	11	com12 13	. . . . 5 |- (-. A e. B -> (A e. suc B -> A = B))
13	9, 12	syl9 55	. . . 4 |- (A e. On -> (-. A e. B -> (suc A = suc B -> A = B)))
14		sucssel 2321	. . . . . 6 |- (B e. On -> (suc B (_ suc A -> B e. suc A))
15		eqimss2 1549	. . . . . 6 |- (suc A = suc B -> suc B (_ suc A)
16	14, 15	syl5 22	. . . . 5 |- (B e. On -> (suc A = suc B -> B e. suc A))
17		elsuci 2289	. . . . . . . 8 |- (B e. suc A -> (B e. A \/ B = A))
18	17	ord 202	. . . . . . 7 |- (B e. suc A -> (-. B e. A -> B = A))
19	18	com12 13	. . . . . 6 |- (-. B e. A -> (B e. suc A -> B = A))
20		cleqcom 1103	. . . . . 6 |- (B = A <-> A = B)
21	19, 20	syl6ib 185	. . . . 5 |- (-. B e. A -> (B e. suc A -> A = B))
22	16, 21	syl9 55	. . . 4 |- (B e. On -> (-. B e. A -> (suc A = suc B -> A = B)))
23	13, 22	jaao 330	. . 3 |- ((A e. On /\ B e. On) -> ((-. A e. B \/ -. B e. A) -> (suc A = suc B -> A = B)))
24	6, 23	mpd 46	. 2 |- ((A e. On /\ B e. On) -> (suc A = suc B -> A = B))
25		suceq 2288	. . 3 |- (A = B -> suc A = suc B)
26	25	a1i 7	. 2 |- ((A e. On /\ B e. On) -> (A = B -> suc A = suc B))
27	24, 26	impbid 397	1 |- ((A e. On /\ B e. On) -> (suc A = suc B <-> A = B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  peano4 2393  limenpsi 3400

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

metamath.org