Theorem dflec2 6210

Description: Cardinal less than or equal in terms of cardinal addition. Theorem XI.2.22 of [Rosser] p. 377. (Contributed by SF, 11-Mar-2015.)

Assertion

Ref	Expression
dflec2	|- ((M e. NC /\ N e. NC ) -> (M <_c N <-> E.p e. NC N = (M +o p)))

Distinct variable groups:   M,p   N,p

Proof of Theorem dflec2

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brlecg 6112	. . 3 |- ((M e. NC /\ N e. NC ) -> (M <_c N <-> E.a e. M E.b e. N a C_ b))
2		ncseqnc 6128	. . . . . . 7 |- (M e. NC -> (M = Nc a <-> a e. M))
3		ncseqnc 6128	. . . . . . 7 |- (N e. NC -> (N = Nc b <-> b e. N))
4	2, 3	bi2anan9 843	. . . . . 6 |- ((M e. NC /\ N e. NC ) -> ((M = Nc a /\ N = Nc b) <-> (a e. M /\ b e. N)))
5	4	biimpar 471	. . . . 5 |- (((M e. NC /\ N e. NC ) /\ (a e. M /\ b e. N)) -> (M = Nc a /\ N = Nc b))
6		vex 2862	. . . . . . . . 9 |- b e. _V
7		vex 2862	. . . . . . . . 9 |- a e. _V
8	6, 7	difex 4107	. . . . . . . 8 |- (b \ a) e. _V
9	8	ncelncsi 6121	. . . . . . 7 |- Nc (b \ a) e. NC
10		disjdif 3622	. . . . . . . . 9 |- (a i^i (b \ a)) = (/)
11	7, 8	ncdisjun 6136	. . . . . . . . 9 |- ((a i^i (b \ a)) = (/) -> Nc (a u. (b \ a)) = ( Nc a +o Nc (b \ a)))
12	10, 11	ax-mp 8	. . . . . . . 8 |- Nc (a u. (b \ a)) = ( Nc a +o Nc (b \ a))
13		undif2 3626	. . . . . . . . . 10 |- (a u. (b \ a)) = (a u. b)
14		ssequn1 3433	. . . . . . . . . . 11 |- (a C_ b <-> (a u. b) = b)
15	14	biimpi 186	. . . . . . . . . 10 |- (a C_ b -> (a u. b) = b)
16	13, 15	syl5eq 2397	. . . . . . . . 9 |- (a C_ b -> (a u. (b \ a)) = b)
17	16	nceqd 6110	. . . . . . . 8 |- (a C_ b -> Nc (a u. (b \ a)) = Nc b)
18	12, 17	syl5reqr 2400	. . . . . . 7 |- (a C_ b -> Nc b = ( Nc a +o Nc (b \ a)))
19		addceq2 4384	. . . . . . . . 9 |- (p = Nc (b \ a) -> ( Nc a +o p) = ( Nc a +o Nc (b \ a)))
20	19	eqeq2d 2364	. . . . . . . 8 |- (p = Nc (b \ a) -> ( Nc b = ( Nc a +o p) <-> Nc b = ( Nc a +o Nc (b \ a))))
21	20	rspcev 2955	. . . . . . 7 |- (( Nc (b \ a) e. NC /\ Nc b = ( Nc a +o Nc (b \ a))) -> E.p e. NC Nc b = ( Nc a +o p))
22	9, 18, 21	sylancr 644	. . . . . 6 |- (a C_ b -> E.p e. NC Nc b = ( Nc a +o p))
23		id 19	. . . . . . . . 9 |- (N = Nc b -> N = Nc b)
24		addceq1 4383	. . . . . . . . 9 |- (M = Nc a -> (M +o p) = ( Nc a +o p))
25	23, 24	eqeqan12d 2368	. . . . . . . 8 |- ((N = Nc b /\ M = Nc a) -> (N = (M +o p) <-> Nc b = ( Nc a +o p)))
26	25	rexbidv 2635	. . . . . . 7 |- ((N = Nc b /\ M = Nc a) -> (E.p e. NC N = (M +o p) <-> E.p e. NC Nc b = ( Nc a +o p)))
27	26	ancoms 439	. . . . . 6 |- ((M = Nc a /\ N = Nc b) -> (E.p e. NC N = (M +o p) <-> E.p e. NC Nc b = ( Nc a +o p)))
28	22, 27	syl5ibr 212	. . . . 5 |- ((M = Nc a /\ N = Nc b) -> (a C_ b -> E.p e. NC N = (M +o p)))
29	5, 28	syl 15	. . . 4 |- (((M e. NC /\ N e. NC ) /\ (a e. M /\ b e. N)) -> (a C_ b -> E.p e. NC N = (M +o p)))
30	29	rexlimdvva 2745	. . 3 |- ((M e. NC /\ N e. NC ) -> (E.a e. M E.b e. N a C_ b -> E.p e. NC N = (M +o p)))
31	1, 30	sylbid 206	. 2 |- ((M e. NC /\ N e. NC ) -> (M <_c N -> E.p e. NC N = (M +o p)))
32		addlecncs 6209	. . . . 5 |- ((M e. NC /\ p e. NC ) -> M <_c (M +o p))
33		breq2 4643	. . . . 5 |- (N = (M +o p) -> (M <_c N <-> M <_c (M +o p)))
34	32, 33	syl5ibrcom 213	. . . 4 |- ((M e. NC /\ p e. NC ) -> (N = (M +o p) -> M <_c N))
35	34	adantlr 695	. . 3 |- (((M e. NC /\ N e. NC ) /\ p e. NC ) -> (N = (M +o p) -> M <_c N))
36	35	rexlimdva 2738	. 2 |- ((M e. NC /\ N e. NC ) -> (E.p e. NC N = (M +o p) -> M <_c N))
37	31, 36	impbid 183	1 |- ((M e. NC /\ N e. NC ) -> (M <_c N <-> E.p e. NC N = (M +o p)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615   \ cdif 3206   u. cun 3207   i^i cin 3208   C_ wss 3257  (/)c0 3550   +o cplc 4375   class class class wbr 4639   NC cncs 6088   <__c clec 6089   Nc cnc 6091

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101

This theorem is referenced by:  lectr  6211  leaddc1  6214  nc0suc  6217  leconnnc  6218  tlecg  6230  letc  6231  nclenn  6249  lemuc1  6253  lecadd2  6266  ncslesuc  6267  nchoicelem14  6302  nchoicelem17  6305