Theorem dflec2 6211

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 ∈ NC ∧ N ∈ NC ) → (M ≤c N ↔ ∃p ∈ NC N = (M +c 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 6113	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c N ↔ ∃a ∈ M ∃b ∈ N a ⊆ b))
2		ncseqnc 6129	. . . . . . 7 ⊢ (M ∈ NC → (M = Nc a ↔ a ∈ M))
3		ncseqnc 6129	. . . . . . 7 ⊢ (N ∈ NC → (N = Nc b ↔ b ∈ N))
4	2, 3	bi2anan9 843	. . . . . 6 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((M = Nc a ∧ N = Nc b) ↔ (a ∈ M ∧ b ∈ N)))
5	4	biimpar 471	. . . . 5 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ (a ∈ M ∧ b ∈ N)) → (M = Nc a ∧ N = Nc b))
6		vex 2863	. . . . . . . . 9 ⊢ b ∈ V
7		vex 2863	. . . . . . . . 9 ⊢ a ∈ V
8	6, 7	difex 4108	. . . . . . . 8 ⊢ (b ∖ a) ∈ V
9	8	ncelncsi 6122	. . . . . . 7 ⊢ Nc (b ∖ a) ∈ NC
10		disjdif 3623	. . . . . . . . 9 ⊢ (a ∩ (b ∖ a)) = ∅
11	7, 8	ncdisjun 6137	. . . . . . . . 9 ⊢ ((a ∩ (b ∖ a)) = ∅ → Nc (a ∪ (b ∖ a)) = ( Nc a +c Nc (b ∖ a)))
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ Nc (a ∪ (b ∖ a)) = ( Nc a +c Nc (b ∖ a))
13		undif2 3627	. . . . . . . . . 10 ⊢ (a ∪ (b ∖ a)) = (a ∪ b)
14		ssequn1 3434	. . . . . . . . . . 11 ⊢ (a ⊆ b ↔ (a ∪ b) = b)
15	14	biimpi 186	. . . . . . . . . 10 ⊢ (a ⊆ b → (a ∪ b) = b)
16	13, 15	syl5eq 2397	. . . . . . . . 9 ⊢ (a ⊆ b → (a ∪ (b ∖ a)) = b)
17	16	nceqd 6111	. . . . . . . 8 ⊢ (a ⊆ b → Nc (a ∪ (b ∖ a)) = Nc b)
18	12, 17	syl5reqr 2400	. . . . . . 7 ⊢ (a ⊆ b → Nc b = ( Nc a +c Nc (b ∖ a)))
19		addceq2 4385	. . . . . . . . 9 ⊢ (p = Nc (b ∖ a) → ( Nc a +c p) = ( Nc a +c Nc (b ∖ a)))
20	19	eqeq2d 2364	. . . . . . . 8 ⊢ (p = Nc (b ∖ a) → ( Nc b = ( Nc a +c p) ↔ Nc b = ( Nc a +c Nc (b ∖ a))))
21	20	rspcev 2956	. . . . . . 7 ⊢ (( Nc (b ∖ a) ∈ NC ∧ Nc b = ( Nc a +c Nc (b ∖ a))) → ∃p ∈ NC Nc b = ( Nc a +c p))
22	9, 18, 21	sylancr 644	. . . . . 6 ⊢ (a ⊆ b → ∃p ∈ NC Nc b = ( Nc a +c p))
23		id 19	. . . . . . . . 9 ⊢ (N = Nc b → N = Nc b)
24		addceq1 4384	. . . . . . . . 9 ⊢ (M = Nc a → (M +c p) = ( Nc a +c p))
25	23, 24	eqeqan12d 2368	. . . . . . . 8 ⊢ ((N = Nc b ∧ M = Nc a) → (N = (M +c p) ↔ Nc b = ( Nc a +c p)))
26	25	rexbidv 2636	. . . . . . 7 ⊢ ((N = Nc b ∧ M = Nc a) → (∃p ∈ NC N = (M +c p) ↔ ∃p ∈ NC Nc b = ( Nc a +c p)))
27	26	ancoms 439	. . . . . 6 ⊢ ((M = Nc a ∧ N = Nc b) → (∃p ∈ NC N = (M +c p) ↔ ∃p ∈ NC Nc b = ( Nc a +c p)))
28	22, 27	syl5ibr 212	. . . . 5 ⊢ ((M = Nc a ∧ N = Nc b) → (a ⊆ b → ∃p ∈ NC N = (M +c p)))
29	5, 28	syl 15	. . . 4 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ (a ∈ M ∧ b ∈ N)) → (a ⊆ b → ∃p ∈ NC N = (M +c p)))
30	29	rexlimdvva 2746	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (∃a ∈ M ∃b ∈ N a ⊆ b → ∃p ∈ NC N = (M +c p)))
31	1, 30	sylbid 206	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c N → ∃p ∈ NC N = (M +c p)))
32		addlecncs 6210	. . . . 5 ⊢ ((M ∈ NC ∧ p ∈ NC ) → M ≤c (M +c p))
33		breq2 4644	. . . . 5 ⊢ (N = (M +c p) → (M ≤c N ↔ M ≤c (M +c p)))
34	32, 33	syl5ibrcom 213	. . . 4 ⊢ ((M ∈ NC ∧ p ∈ NC ) → (N = (M +c p) → M ≤c N))
35	34	adantlr 695	. . 3 ⊢ (((M ∈ NC ∧ N ∈ NC ) ∧ p ∈ NC ) → (N = (M +c p) → M ≤c N))
36	35	rexlimdva 2739	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (∃p ∈ NC N = (M +c p) → M ≤c N))
37	31, 36	impbid 183	1 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ≤c N ↔ ∃p ∈ NC N = (M +c p)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551   +_c cplc 4376   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090   Nc cnc 6092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102

This theorem is used by:  lectr  6212  leaddc1  6215  nc0suc  6218  leconnnc  6219  tlecg  6231  letc  6232  nclenn  6250  lemuc1  6254  lecadd2  6267  ncslesuc  6268  nchoicelem14  6303  nchoicelem17  6306