Theorem suc11 3088

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 2953	. . . . 5 |- (A e. On -> Ord A)
2		ordn2lp 2963	. . . . . 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 3065	. . . . . 6 |- (A e. On -> (suc A (_ suc B -> A e. suc B))
8		eqimss 2105	. . . . . 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 3030	. . . . . . 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 3065	. . . . . 6 |- (B e. On -> (suc B (_ suc A -> B e. suc A))
15		eqimss2 2106	. . . . . 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 3030	. . . . . . . 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 1474	. . . . . 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 3029	. 2 |- (A = B -> suc A = suc B)
26	24, 25	impbid1 516	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 954   e. wcel 956   (_ wss 2043  Ord word 2942  Oncon0 2943  suc csuc 2945

This theorem is referenced by:  peano4 3147  limenpsi 4491

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949

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