Theorem nnaordex 7718

Description: Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnaordex	|- ( ( A e. om /\ B e. om ) -> ( A e. B <-> E. x e. om ( (/) e. x /\ ( A +o x ) = B ) ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem nnaordex

Step	Hyp	Ref	Expression
1		nnon 7071	. . . . . 6 |- ( B e. om -> B e. On )
2	1	adantl 482	. . . . 5 |- ( ( A e. om /\ B e. om ) -> B e. On )
3		onelss 5766	. . . . 5 |- ( B e. On -> ( A e. B -> A C_ B ) )
4	2, 3	syl 17	. . . 4 |- ( ( A e. om /\ B e. om ) -> ( A e. B -> A C_ B ) )
5		nnawordex 7717	. . . 4 |- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> E. x e. om ( A +o x ) = B ) )
6	4, 5	sylibd 229	. . 3 |- ( ( A e. om /\ B e. om ) -> ( A e. B -> E. x e. om ( A +o x ) = B ) )
7		simplr 792	. . . . . . . . 9 |- ( ( ( A e. om /\ A e. B ) /\ x e. om ) -> A e. B )
8		eleq2 2690	. . . . . . . . 9 |- ( ( A +o x ) = B -> ( A e. ( A +o x ) <-> A e. B ) )
9	7, 8	syl5ibrcom 237	. . . . . . . 8 |- ( ( ( A e. om /\ A e. B ) /\ x e. om ) -> ( ( A +o x ) = B -> A e. ( A +o x ) ) )
10		peano1 7085	. . . . . . . . . . . 12 |- (/) e. om
11		nnaord 7699	. . . . . . . . . . . 12 |- ( ( (/) e. om /\ x e. om /\ A e. om ) -> ( (/) e. x <-> ( A +o (/) ) e. ( A +o x ) ) )
12	10, 11	mp3an1 1411	. . . . . . . . . . 11 |- ( ( x e. om /\ A e. om ) -> ( (/) e. x <-> ( A +o (/) ) e. ( A +o x ) ) )
13	12	ancoms 469	. . . . . . . . . 10 |- ( ( A e. om /\ x e. om ) -> ( (/) e. x <-> ( A +o (/) ) e. ( A +o x ) ) )
14		nna0 7684	. . . . . . . . . . . 12 |- ( A e. om -> ( A +o (/) ) = A )
15	14	adantr 481	. . . . . . . . . . 11 |- ( ( A e. om /\ x e. om ) -> ( A +o (/) ) = A )
16	15	eleq1d 2686	. . . . . . . . . 10 |- ( ( A e. om /\ x e. om ) -> ( ( A +o (/) ) e. ( A +o x ) <-> A e. ( A +o x ) ) )
17	13, 16	bitrd 268	. . . . . . . . 9 |- ( ( A e. om /\ x e. om ) -> ( (/) e. x <-> A e. ( A +o x ) ) )
18	17	adantlr 751	. . . . . . . 8 |- ( ( ( A e. om /\ A e. B ) /\ x e. om ) -> ( (/) e. x <-> A e. ( A +o x ) ) )
19	9, 18	sylibrd 249	. . . . . . 7 |- ( ( ( A e. om /\ A e. B ) /\ x e. om ) -> ( ( A +o x ) = B -> (/) e. x ) )
20	19	ancrd 577	. . . . . 6 |- ( ( ( A e. om /\ A e. B ) /\ x e. om ) -> ( ( A +o x ) = B -> ( (/) e. x /\ ( A +o x ) = B ) ) )
21	20	reximdva 3017	. . . . 5 |- ( ( A e. om /\ A e. B ) -> ( E. x e. om ( A +o x ) = B -> E. x e. om ( (/) e. x /\ ( A +o x ) = B ) ) )
22	21	ex 450	. . . 4 |- ( A e. om -> ( A e. B -> ( E. x e. om ( A +o x ) = B -> E. x e. om ( (/) e. x /\ ( A +o x ) = B ) ) ) )
23	22	adantr 481	. . 3 |- ( ( A e. om /\ B e. om ) -> ( A e. B -> ( E. x e. om ( A +o x ) = B -> E. x e. om ( (/) e. x /\ ( A +o x ) = B ) ) ) )
24	6, 23	mpdd 43	. 2 |- ( ( A e. om /\ B e. om ) -> ( A e. B -> E. x e. om ( (/) e. x /\ ( A +o x ) = B ) ) )
25	17	biimpa 501	. . . . . 6 |- ( ( ( A e. om /\ x e. om ) /\ (/) e. x ) -> A e. ( A +o x ) )
26	25, 8	syl5ibcom 235	. . . . 5 |- ( ( ( A e. om /\ x e. om ) /\ (/) e. x ) -> ( ( A +o x ) = B -> A e. B ) )
27	26	expimpd 629	. . . 4 |- ( ( A e. om /\ x e. om ) -> ( ( (/) e. x /\ ( A +o x ) = B ) -> A e. B ) )
28	27	rexlimdva 3031	. . 3 |- ( A e. om -> ( E. x e. om ( (/) e. x /\ ( A +o x ) = B ) -> A e. B ) )
29	28	adantr 481	. 2 |- ( ( A e. om /\ B e. om ) -> ( E. x e. om ( (/) e. x /\ ( A +o x ) = B ) -> A e. B ) )
30	24, 29	impbid 202	1 |- ( ( A e. om /\ B e. om ) -> ( A e. B <-> E. x e. om ( (/) e. x /\ ( A +o x ) = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   (/)c0 3915   Oncon0 5723  (class class class)co 6650   omcom 7065   +o coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  ltexpi  9724