Theorem nclenn 6250

Description: A cardinal that is less than or equal to a natural is a natural. Theorem XI.3.3 of [Rosser] p. 391. (Contributed by SF, 19-Mar-2015.)

Assertion

Ref	Expression
nclenn	⊢ ((M ∈ NC ∧ N ∈ Nn ∧ M ≤c N) → M ∈ Nn )

Proof of Theorem nclenn

Dummy variables m n p q x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nclennlem1 6249	. . . . 5 ⊢ {x ∣ ∀n ∈ NC (n ≤c x → n ∈ Nn )} ∈ V
2		breq2 4644	. . . . . . 7 ⊢ (x = 0c → (n ≤c x ↔ n ≤c 0c))
3	2	imbi1d 308	. . . . . 6 ⊢ (x = 0c → ((n ≤c x → n ∈ Nn ) ↔ (n ≤c 0c → n ∈ Nn )))
4	3	ralbidv 2635	. . . . 5 ⊢ (x = 0c → (∀n ∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c 0c → n ∈ Nn )))
5		breq2 4644	. . . . . . 7 ⊢ (x = m → (n ≤c x ↔ n ≤c m))
6	5	imbi1d 308	. . . . . 6 ⊢ (x = m → ((n ≤c x → n ∈ Nn ) ↔ (n ≤c m → n ∈ Nn )))
7	6	ralbidv 2635	. . . . 5 ⊢ (x = m → (∀n ∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c m → n ∈ Nn )))
8		breq2 4644	. . . . . . 7 ⊢ (x = (m +c 1c) → (n ≤c x ↔ n ≤c (m +c 1c)))
9	8	imbi1d 308	. . . . . 6 ⊢ (x = (m +c 1c) → ((n ≤c x → n ∈ Nn ) ↔ (n ≤c (m +c 1c) → n ∈ Nn )))
10	9	ralbidv 2635	. . . . 5 ⊢ (x = (m +c 1c) → (∀n ∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c (m +c 1c) → n ∈ Nn )))
11		breq2 4644	. . . . . . 7 ⊢ (x = N → (n ≤c x ↔ n ≤c N))
12	11	imbi1d 308	. . . . . 6 ⊢ (x = N → ((n ≤c x → n ∈ Nn ) ↔ (n ≤c N → n ∈ Nn )))
13	12	ralbidv 2635	. . . . 5 ⊢ (x = N → (∀n ∈ NC (n ≤c x → n ∈ Nn ) ↔ ∀n ∈ NC (n ≤c N → n ∈ Nn )))
14		le0nc 6201	. . . . . . 7 ⊢ (n ∈ NC → 0c ≤c n)
15		0cnc 6139	. . . . . . . . . . 11 ⊢ 0c ∈ NC
16		sbth 6207	. . . . . . . . . . 11 ⊢ ((n ∈ NC ∧ 0c ∈ NC ) → ((n ≤c 0c ∧ 0c ≤c n) → n = 0c))
17	15, 16	mpan2 652	. . . . . . . . . 10 ⊢ (n ∈ NC → ((n ≤c 0c ∧ 0c ≤c n) → n = 0c))
18	17	imp 418	. . . . . . . . 9 ⊢ ((n ∈ NC ∧ (n ≤c 0c ∧ 0c ≤c n)) → n = 0c)
19		peano1 4403	. . . . . . . . 9 ⊢ 0c ∈ Nn
20	18, 19	syl6eqel 2441	. . . . . . . 8 ⊢ ((n ∈ NC ∧ (n ≤c 0c ∧ 0c ≤c n)) → n ∈ Nn )
21	20	ex 423	. . . . . . 7 ⊢ (n ∈ NC → ((n ≤c 0c ∧ 0c ≤c n) → n ∈ Nn ))
22	14, 21	mpan2d 655	. . . . . 6 ⊢ (n ∈ NC → (n ≤c 0c → n ∈ Nn ))
23	22	rgen 2680	. . . . 5 ⊢ ∀n ∈ NC (n ≤c 0c → n ∈ Nn )
24		peano2 4404	. . . . . . . . . . . 12 ⊢ (m ∈ Nn → (m +c 1c) ∈ Nn )
25		nnnc 6147	. . . . . . . . . . . 12 ⊢ ((m +c 1c) ∈ Nn → (m +c 1c) ∈ NC )
26	24, 25	syl 15	. . . . . . . . . . 11 ⊢ (m ∈ Nn → (m +c 1c) ∈ NC )
27		dflec2 6211	. . . . . . . . . . 11 ⊢ ((n ∈ NC ∧ (m +c 1c) ∈ NC ) → (n ≤c (m +c 1c) ↔ ∃p ∈ NC (m +c 1c) = (n +c p)))
28	26, 27	sylan2 460	. . . . . . . . . 10 ⊢ ((n ∈ NC ∧ m ∈ Nn ) → (n ≤c (m +c 1c) ↔ ∃p ∈ NC (m +c 1c) = (n +c p)))
29	28	ancoms 439	. . . . . . . . 9 ⊢ ((m ∈ Nn ∧ n ∈ NC ) → (n ≤c (m +c 1c) ↔ ∃p ∈ NC (m +c 1c) = (n +c p)))
30	29	3adant3 975	. . . . . . . 8 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (n ≤c (m +c 1c) ↔ ∃p ∈ NC (m +c 1c) = (n +c p)))
31		nc0suc 6218	. . . . . . . . . 10 ⊢ (p ∈ NC → (p = 0c ∨ ∃q ∈ NC p = (q +c 1c)))
32		addceq2 4385	. . . . . . . . . . . . . . . . . 18 ⊢ (p = 0c → (n +c p) = (n +c 0c))
33		addcid1 4406	. . . . . . . . . . . . . . . . . 18 ⊢ (n +c 0c) = n
34	32, 33	syl6eq 2401	. . . . . . . . . . . . . . . . 17 ⊢ (p = 0c → (n +c p) = n)
35	34	eqeq2d 2364	. . . . . . . . . . . . . . . 16 ⊢ (p = 0c → ((m +c 1c) = (n +c p) ↔ (m +c 1c) = n))
36	35	biimpa 470	. . . . . . . . . . . . . . 15 ⊢ ((p = 0c ∧ (m +c 1c) = (n +c p)) → (m +c 1c) = n)
37		eleq1 2413	. . . . . . . . . . . . . . . 16 ⊢ ((m +c 1c) = n → ((m +c 1c) ∈ Nn ↔ n ∈ Nn ))
38	37	biimpcd 215	. . . . . . . . . . . . . . 15 ⊢ ((m +c 1c) ∈ Nn → ((m +c 1c) = n → n ∈ Nn ))
39	36, 38	syl5 28	. . . . . . . . . . . . . 14 ⊢ ((m +c 1c) ∈ Nn → ((p = 0c ∧ (m +c 1c) = (n +c p)) → n ∈ Nn ))
40	39	exp3a 425	. . . . . . . . . . . . 13 ⊢ ((m +c 1c) ∈ Nn → (p = 0c → ((m +c 1c) = (n +c p) → n ∈ Nn )))
41	24, 40	syl 15	. . . . . . . . . . . 12 ⊢ (m ∈ Nn → (p = 0c → ((m +c 1c) = (n +c p) → n ∈ Nn )))
42	41	3ad2ant1 976	. . . . . . . . . . 11 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (p = 0c → ((m +c 1c) = (n +c p) → n ∈ Nn )))
43		addceq2 4385	. . . . . . . . . . . . . . . . 17 ⊢ (p = (q +c 1c) → (n +c p) = (n +c (q +c 1c)))
44		addcass 4416	. . . . . . . . . . . . . . . . 17 ⊢ ((n +c q) +c 1c) = (n +c (q +c 1c))
45	43, 44	syl6eqr 2403	. . . . . . . . . . . . . . . 16 ⊢ (p = (q +c 1c) → (n +c p) = ((n +c q) +c 1c))
46	45	eqeq2d 2364	. . . . . . . . . . . . . . 15 ⊢ (p = (q +c 1c) → ((m +c 1c) = (n +c p) ↔ (m +c 1c) = ((n +c q) +c 1c)))
47	46	biimpa 470	. . . . . . . . . . . . . 14 ⊢ ((p = (q +c 1c) ∧ (m +c 1c) = (n +c p)) → (m +c 1c) = ((n +c q) +c 1c))
48		nnnc 6147	. . . . . . . . . . . . . . . . . 18 ⊢ (m ∈ Nn → m ∈ NC )
49	48	3ad2ant1 976	. . . . . . . . . . . . . . . . 17 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → m ∈ NC )
50	49	adantr 451	. . . . . . . . . . . . . . . 16 ⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → m ∈ NC )
51		ncaddccl 6145	. . . . . . . . . . . . . . . . 17 ⊢ ((n ∈ NC ∧ q ∈ NC ) → (n +c q) ∈ NC )
52	51	3ad2antl2 1118	. . . . . . . . . . . . . . . 16 ⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → (n +c q) ∈ NC )
53		peano4nc 6151	. . . . . . . . . . . . . . . 16 ⊢ ((m ∈ NC ∧ (n +c q) ∈ NC ) → ((m +c 1c) = ((n +c q) +c 1c) ↔ m = (n +c q)))
54	50, 52, 53	syl2anc 642	. . . . . . . . . . . . . . 15 ⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → ((m +c 1c) = ((n +c q) +c 1c) ↔ m = (n +c q)))
55		addlecncs 6210	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((n ∈ NC ∧ q ∈ NC ) → n ≤c (n +c q))
56		breq2 4644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (m = (n +c q) → (n ≤c m ↔ n ≤c (n +c q)))
57	55, 56	syl5ibrcom 213	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((n ∈ NC ∧ q ∈ NC ) → (m = (n +c q) → n ≤c m))
58	57	ex 423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (n ∈ NC → (q ∈ NC → (m = (n +c q) → n ≤c m)))
59	58	com23 72	. . . . . . . . . . . . . . . . . . . 20 ⊢ (n ∈ NC → (m = (n +c q) → (q ∈ NC → n ≤c m)))
60	59	adantl 452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((m ∈ Nn ∧ n ∈ NC ) → (m = (n +c q) → (q ∈ NC → n ≤c m)))
61		pm2.27 35	. . . . . . . . . . . . . . . . . . 19 ⊢ (n ≤c m → ((n ≤c m → n ∈ Nn ) → n ∈ Nn ))
62	60, 61	syl8 65	. . . . . . . . . . . . . . . . . 18 ⊢ ((m ∈ Nn ∧ n ∈ NC ) → (m = (n +c q) → (q ∈ NC → ((n ≤c m → n ∈ Nn ) → n ∈ Nn ))))
63	62	com24 81	. . . . . . . . . . . . . . . . 17 ⊢ ((m ∈ Nn ∧ n ∈ NC ) → ((n ≤c m → n ∈ Nn ) → (q ∈ NC → (m = (n +c q) → n ∈ Nn ))))
64	63	3impia 1148	. . . . . . . . . . . . . . . 16 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (q ∈ NC → (m = (n +c q) → n ∈ Nn )))
65	64	imp 418	. . . . . . . . . . . . . . 15 ⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → (m = (n +c q) → n ∈ Nn ))
66	54, 65	sylbid 206	. . . . . . . . . . . . . 14 ⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → ((m +c 1c) = ((n +c q) +c 1c) → n ∈ Nn ))
67	47, 66	syl5 28	. . . . . . . . . . . . 13 ⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → ((p = (q +c 1c) ∧ (m +c 1c) = (n +c p)) → n ∈ Nn ))
68	67	exp3a 425	. . . . . . . . . . . 12 ⊢ (((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) ∧ q ∈ NC ) → (p = (q +c 1c) → ((m +c 1c) = (n +c p) → n ∈ Nn )))
69	68	rexlimdva 2739	. . . . . . . . . . 11 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (∃q ∈ NC p = (q +c 1c) → ((m +c 1c) = (n +c p) → n ∈ Nn )))
70	42, 69	jaod 369	. . . . . . . . . 10 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → ((p = 0c ∨ ∃q ∈ NC p = (q +c 1c)) → ((m +c 1c) = (n +c p) → n ∈ Nn )))
71	31, 70	syl5 28	. . . . . . . . 9 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (p ∈ NC → ((m +c 1c) = (n +c p) → n ∈ Nn )))
72	71	rexlimdv 2738	. . . . . . . 8 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (∃p ∈ NC (m +c 1c) = (n +c p) → n ∈ Nn ))
73	30, 72	sylbid 206	. . . . . . 7 ⊢ ((m ∈ Nn ∧ n ∈ NC ∧ (n ≤c m → n ∈ Nn )) → (n ≤c (m +c 1c) → n ∈ Nn ))
74	73	3expia 1153	. . . . . 6 ⊢ ((m ∈ Nn ∧ n ∈ NC ) → ((n ≤c m → n ∈ Nn ) → (n ≤c (m +c 1c) → n ∈ Nn )))
75	74	ralimdva 2693	. . . . 5 ⊢ (m ∈ Nn → (∀n ∈ NC (n ≤c m → n ∈ Nn ) → ∀n ∈ NC (n ≤c (m +c 1c) → n ∈ Nn )))
76	1, 4, 7, 10, 13, 23, 75	finds 4412	. . . 4 ⊢ (N ∈ Nn → ∀n ∈ NC (n ≤c N → n ∈ Nn ))
77		breq1 4643	. . . . . 6 ⊢ (n = M → (n ≤c N ↔ M ≤c N))
78		eleq1 2413	. . . . . 6 ⊢ (n = M → (n ∈ Nn ↔ M ∈ Nn ))
79	77, 78	imbi12d 311	. . . . 5 ⊢ (n = M → ((n ≤c N → n ∈ Nn ) ↔ (M ≤c N → M ∈ Nn )))
80	79	rspccv 2953	. . . 4 ⊢ (∀n ∈ NC (n ≤c N → n ∈ Nn ) → (M ∈ NC → (M ≤c N → M ∈ Nn )))
81	76, 80	syl 15	. . 3 ⊢ (N ∈ Nn → (M ∈ NC → (M ≤c N → M ∈ Nn )))
82	81	com12 27	. 2 ⊢ (M ∈ NC → (N ∈ Nn → (M ≤c N → M ∈ Nn )))
83	82	3imp 1145	1 ⊢ ((M ∈ NC ∧ N ∈ Nn ∧ M ≤c N) → M ∈ Nn )

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090

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-fv 4796  df-2nd 4798  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-clos1 5874  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:  nchoicelem17  6306