Theorem suc11g 4699

Description: The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)

Assertion

Ref	Expression
suc11g	|- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )

Proof of Theorem suc11g

Step	Hyp	Ref	Expression
1		en2lp 4696	. . . 4 |- -. ( B e. A /\ A e. B )
2		sucidg 4556	. . . . . . . . . . . 12 |- ( B e. W -> B e. suc B )
3		eleq2 2302	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
4	2, 3	syl5ibrcom 157	. . . . . . . . . . 11 |- ( B e. W -> ( suc A = suc B -> B e. suc A ) )
5		elsucg 4544	. . . . . . . . . . 11 |- ( B e. W -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
6	4, 5	sylibd 149	. . . . . . . . . 10 |- ( B e. W -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
7	6	imp 124	. . . . . . . . 9 |- ( ( B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
8	7	3adant1 1046	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
9		sucidg 4556	. . . . . . . . . . . 12 |- ( A e. V -> A e. suc A )
10		eleq2 2302	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
11	9, 10	syl5ibcom 155	. . . . . . . . . . 11 |- ( A e. V -> ( suc A = suc B -> A e. suc B ) )
12		elsucg 4544	. . . . . . . . . . 11 |- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
13	11, 12	sylibd 149	. . . . . . . . . 10 |- ( A e. V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
14	13	imp 124	. . . . . . . . 9 |- ( ( A e. V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
15	14	3adant2 1047	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
16	8, 15	jca 306	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) )
17		eqcom 2240	. . . . . . . . 9 |- ( B = A <-> A = B )
18	17	orbi2i 774	. . . . . . . 8 |- ( ( B e. A \/ B = A ) <-> ( B e. A \/ A = B ) )
19	18	anbi1i 462	. . . . . . 7 |- ( ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
20	16, 19	sylib 122	. . . . . 6 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
21		ordir 829	. . . . . 6 |- ( ( ( B e. A /\ A e. B ) \/ A = B ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
22	20, 21	sylibr 134	. . . . 5 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A /\ A e. B ) \/ A = B ) )
23	22	ord 736	. . . 4 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( -. ( B e. A /\ A e. B ) -> A = B ) )
24	1, 23	mpi 15	. . 3 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> A = B )
25	24	3expia 1236	. 2 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B -> A = B ) )
26		suceq 4542	. 2 |- ( A = B -> suc A = suc B )
27	25, 26	impbid1 142	1 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   suc csuc 4505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-sn 3711  df-pr 3712  df-suc 4511

This theorem is referenced by:  suc11  4700  peano4  4739  frecsuclem  6667