Theorem nnawordex 6696

Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem nnawordex

Step	Hyp	Ref	Expression
1		nntri3or 6660	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( A e. B \/ A = B \/ B e. A ) )
2	1	3adant3 1043	. . . 4 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> ( A e. B \/ A = B \/ B e. A ) )
3		nnaordex 6695	. . . . . . 7 |- ( ( A e. om /\ B e. om ) -> ( A e. B <-> E. x e. om ( (/) e. x /\ ( A +o x ) = B ) ) )
4		simpr 110	. . . . . . . 8 |- ( ( (/) e. x /\ ( A +o x ) = B ) -> ( A +o x ) = B )
5	4	reximi 2629	. . . . . . 7 |- ( E. x e. om ( (/) e. x /\ ( A +o x ) = B ) -> E. x e. om ( A +o x ) = B )
6	3, 5	biimtrdi 163	. . . . . 6 |- ( ( A e. om /\ B e. om ) -> ( A e. B -> E. x e. om ( A +o x ) = B ) )
7	6	3adant3 1043	. . . . 5 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> ( A e. B -> E. x e. om ( A +o x ) = B ) )
8		nna0 6641	. . . . . . . 8 |- ( A e. om -> ( A +o (/) ) = A )
9	8	3ad2ant1 1044	. . . . . . 7 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> ( A +o (/) ) = A )
10		eqeq2 2241	. . . . . . 7 |- ( A = B -> ( ( A +o (/) ) = A <-> ( A +o (/) ) = B ) )
11	9, 10	syl5ibcom 155	. . . . . 6 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> ( A = B -> ( A +o (/) ) = B ) )
12		peano1 4692	. . . . . . 7 |- (/) e. om
13		oveq2 6025	. . . . . . . . 9 |- ( x = (/) -> ( A +o x ) = ( A +o (/) ) )
14	13	eqeq1d 2240	. . . . . . . 8 |- ( x = (/) -> ( ( A +o x ) = B <-> ( A +o (/) ) = B ) )
15	14	rspcev 2910	. . . . . . 7 |- ( ( (/) e. om /\ ( A +o (/) ) = B ) -> E. x e. om ( A +o x ) = B )
16	12, 15	mpan 424	. . . . . 6 |- ( ( A +o (/) ) = B -> E. x e. om ( A +o x ) = B )
17	11, 16	syl6 33	. . . . 5 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> ( A = B -> E. x e. om ( A +o x ) = B ) )
18		nntri1 6663	. . . . . . 7 |- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> -. B e. A ) )
19	18	biimp3a 1381	. . . . . 6 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> -. B e. A )
20	19	pm2.21d 624	. . . . 5 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> ( B e. A -> E. x e. om ( A +o x ) = B ) )
21	7, 17, 20	3jaod 1340	. . . 4 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> ( ( A e. B \/ A = B \/ B e. A ) -> E. x e. om ( A +o x ) = B ) )
22	2, 21	mpd 13	. . 3 |- ( ( A e. om /\ B e. om /\ A C_ B ) -> E. x e. om ( A +o x ) = B )
23	22	3expia 1231	. 2 |- ( ( A e. om /\ B e. om ) -> ( A C_ B -> E. x e. om ( A +o x ) = B ) )
24		nnaword1 6680	. . . . 5 |- ( ( A e. om /\ x e. om ) -> A C_ ( A +o x ) )
25		sseq2 3251	. . . . 5 |- ( ( A +o x ) = B -> ( A C_ ( A +o x ) <-> A C_ B ) )
26	24, 25	syl5ibcom 155	. . . 4 |- ( ( A e. om /\ x e. om ) -> ( ( A +o x ) = B -> A C_ B ) )
27	26	rexlimdva 2650	. . 3 |- ( A e. om -> ( E. x e. om ( A +o x ) = B -> A C_ B ) )
28	27	adantr 276	. 2 |- ( ( A e. om /\ B e. om ) -> ( E. x e. om ( A +o x ) = B -> A C_ B ) )
29	23, 28	impbid 129	1 |- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> E. x e. om ( A +o x ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1003   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   C_ wss 3200   (/)c0 3494   omcom 4688  (class class class)co 6017   +o coa 6578

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-oadd 6585

This theorem is referenced by:  prarloclemn  7718