Theorem leconnnc 6218

Description: Cardinal less than or equal is total over the naturals. (Contributed by SF, 12-Mar-2015.)

Assertion

Ref	Expression
leconnnc	⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (A ≤c B ∨ B ≤c A))

Proof of Theorem leconnnc

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

Step	Hyp	Ref	Expression
1		breq2 4643	. . . . . 6 ⊢ (n = B → (A ≤c n ↔ A ≤c B))
2		breq1 4642	. . . . . 6 ⊢ (n = B → (n ≤c A ↔ B ≤c A))
3	1, 2	orbi12d 690	. . . . 5 ⊢ (n = B → ((A ≤c n ∨ n ≤c A) ↔ (A ≤c B ∨ B ≤c A)))
4	3	imbi2d 307	. . . 4 ⊢ (n = B → ((A ∈ Nn → (A ≤c n ∨ n ≤c A)) ↔ (A ∈ Nn → (A ≤c B ∨ B ≤c A))))
5		elun 3220	. . . . . . . . . . . 12 ⊢ (a ∈ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n})) ↔ (a ∈ (◡ ≤c “ {n}) ∨ a ∈ ( ≤c “ {n})))
6		eliniseg 5020	. . . . . . . . . . . . 13 ⊢ (a ∈ (◡ ≤c “ {n}) ↔ a ≤c n)
7		elimasn 5019	. . . . . . . . . . . . . 14 ⊢ (a ∈ ( ≤c “ {n}) ↔ ⟨n, a⟩ ∈ ≤c )
8		df-br 4640	. . . . . . . . . . . . . 14 ⊢ (n ≤c a ↔ ⟨n, a⟩ ∈ ≤c )
9	7, 8	bitr4i 243	. . . . . . . . . . . . 13 ⊢ (a ∈ ( ≤c “ {n}) ↔ n ≤c a)
10	6, 9	orbi12i 507	. . . . . . . . . . . 12 ⊢ ((a ∈ (◡ ≤c “ {n}) ∨ a ∈ ( ≤c “ {n})) ↔ (a ≤c n ∨ n ≤c a))
11	5, 10	bitri 240	. . . . . . . . . . 11 ⊢ (a ∈ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n})) ↔ (a ≤c n ∨ n ≤c a))
12	11	abbi2i 2464	. . . . . . . . . 10 ⊢ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n})) = {a ∣ (a ≤c n ∨ n ≤c a)}
13	12	uneq2i 3415	. . . . . . . . 9 ⊢ ({a ∣ ¬ n ∈ Nn } ∪ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n}))) = ({a ∣ ¬ n ∈ Nn } ∪ {a ∣ (a ≤c n ∨ n ≤c a)})
14		unab 3521	. . . . . . . . 9 ⊢ ({a ∣ ¬ n ∈ Nn } ∪ {a ∣ (a ≤c n ∨ n ≤c a)}) = {a ∣ (¬ n ∈ Nn ∨ (a ≤c n ∨ n ≤c a))}
15	13, 14	eqtri 2373	. . . . . . . 8 ⊢ ({a ∣ ¬ n ∈ Nn } ∪ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n}))) = {a ∣ (¬ n ∈ Nn ∨ (a ≤c n ∨ n ≤c a))}
16		imor 401	. . . . . . . . 9 ⊢ ((n ∈ Nn → (a ≤c n ∨ n ≤c a)) ↔ (¬ n ∈ Nn ∨ (a ≤c n ∨ n ≤c a)))
17	16	abbii 2465	. . . . . . . 8 ⊢ {a ∣ (n ∈ Nn → (a ≤c n ∨ n ≤c a))} = {a ∣ (¬ n ∈ Nn ∨ (a ≤c n ∨ n ≤c a))}
18	15, 17	eqtr4i 2376	. . . . . . 7 ⊢ ({a ∣ ¬ n ∈ Nn } ∪ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n}))) = {a ∣ (n ∈ Nn → (a ≤c n ∨ n ≤c a))}
19		abexv 4324	. . . . . . . 8 ⊢ {a ∣ ¬ n ∈ Nn } ∈ V
20		lecex 6115	. . . . . . . . . . 11 ⊢ ≤c ∈ V
21	20	cnvex 5102	. . . . . . . . . 10 ⊢ ◡ ≤c ∈ V
22		snex 4111	. . . . . . . . . 10 ⊢ {n} ∈ V
23	21, 22	imaex 4747	. . . . . . . . 9 ⊢ (◡ ≤c “ {n}) ∈ V
24	20, 22	imaex 4747	. . . . . . . . 9 ⊢ ( ≤c “ {n}) ∈ V
25	23, 24	unex 4106	. . . . . . . 8 ⊢ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n})) ∈ V
26	19, 25	unex 4106	. . . . . . 7 ⊢ ({a ∣ ¬ n ∈ Nn } ∪ ((◡ ≤c “ {n}) ∪ ( ≤c “ {n}))) ∈ V
27	18, 26	eqeltrri 2424	. . . . . 6 ⊢ {a ∣ (n ∈ Nn → (a ≤c n ∨ n ≤c a))} ∈ V
28		breq1 4642	. . . . . . . 8 ⊢ (a = 0c → (a ≤c n ↔ 0c ≤c n))
29		breq2 4643	. . . . . . . 8 ⊢ (a = 0c → (n ≤c a ↔ n ≤c 0c))
30	28, 29	orbi12d 690	. . . . . . 7 ⊢ (a = 0c → ((a ≤c n ∨ n ≤c a) ↔ (0c ≤c n ∨ n ≤c 0c)))
31	30	imbi2d 307	. . . . . 6 ⊢ (a = 0c → ((n ∈ Nn → (a ≤c n ∨ n ≤c a)) ↔ (n ∈ Nn → (0c ≤c n ∨ n ≤c 0c))))
32		breq1 4642	. . . . . . . 8 ⊢ (a = m → (a ≤c n ↔ m ≤c n))
33		breq2 4643	. . . . . . . 8 ⊢ (a = m → (n ≤c a ↔ n ≤c m))
34	32, 33	orbi12d 690	. . . . . . 7 ⊢ (a = m → ((a ≤c n ∨ n ≤c a) ↔ (m ≤c n ∨ n ≤c m)))
35	34	imbi2d 307	. . . . . 6 ⊢ (a = m → ((n ∈ Nn → (a ≤c n ∨ n ≤c a)) ↔ (n ∈ Nn → (m ≤c n ∨ n ≤c m))))
36		breq1 4642	. . . . . . . 8 ⊢ (a = (m +c 1c) → (a ≤c n ↔ (m +c 1c) ≤c n))
37		breq2 4643	. . . . . . . 8 ⊢ (a = (m +c 1c) → (n ≤c a ↔ n ≤c (m +c 1c)))
38	36, 37	orbi12d 690	. . . . . . 7 ⊢ (a = (m +c 1c) → ((a ≤c n ∨ n ≤c a) ↔ ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
39	38	imbi2d 307	. . . . . 6 ⊢ (a = (m +c 1c) → ((n ∈ Nn → (a ≤c n ∨ n ≤c a)) ↔ (n ∈ Nn → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c)))))
40		breq1 4642	. . . . . . . 8 ⊢ (a = A → (a ≤c n ↔ A ≤c n))
41		breq2 4643	. . . . . . . 8 ⊢ (a = A → (n ≤c a ↔ n ≤c A))
42	40, 41	orbi12d 690	. . . . . . 7 ⊢ (a = A → ((a ≤c n ∨ n ≤c a) ↔ (A ≤c n ∨ n ≤c A)))
43	42	imbi2d 307	. . . . . 6 ⊢ (a = A → ((n ∈ Nn → (a ≤c n ∨ n ≤c a)) ↔ (n ∈ Nn → (A ≤c n ∨ n ≤c A))))
44		nnnc 6146	. . . . . . . 8 ⊢ (n ∈ Nn → n ∈ NC )
45		le0nc 6200	. . . . . . . 8 ⊢ (n ∈ NC → 0c ≤c n)
46	44, 45	syl 15	. . . . . . 7 ⊢ (n ∈ Nn → 0c ≤c n)
47		orc 374	. . . . . . 7 ⊢ (0c ≤c n → (0c ≤c n ∨ n ≤c 0c))
48	46, 47	syl 15	. . . . . 6 ⊢ (n ∈ Nn → (0c ≤c n ∨ n ≤c 0c))
49		nnnc 6146	. . . . . . . . 9 ⊢ (m ∈ Nn → m ∈ NC )
50		dflec2 6210	. . . . . . . . . . 11 ⊢ ((m ∈ NC ∧ n ∈ NC ) → (m ≤c n ↔ ∃p ∈ NC n = (m +c p)))
51		nc0le1 6216	. . . . . . . . . . . . . . . . . 18 ⊢ (p ∈ NC → (p = 0c ∨ 1c ≤c p))
52		1cnc 6139	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1c ∈ NC
53		le0nc 6200	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1c ∈ NC → 0c ≤c 1c)
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0c ≤c 1c
55		breq1 4642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (p = 0c → (p ≤c 1c ↔ 0c ≤c 1c))
56	54, 55	mpbiri 224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (p = 0c → p ≤c 1c)
57	56	orim1i 503	. . . . . . . . . . . . . . . . . . 19 ⊢ ((p = 0c ∨ 1c ≤c p) → (p ≤c 1c ∨ 1c ≤c p))
58	57	a1i 10	. . . . . . . . . . . . . . . . . 18 ⊢ (p ∈ NC → ((p = 0c ∨ 1c ≤c p) → (p ≤c 1c ∨ 1c ≤c p)))
59	51, 58	mpd 14	. . . . . . . . . . . . . . . . 17 ⊢ (p ∈ NC → (p ≤c 1c ∨ 1c ≤c p))
60	59	orcomd 377	. . . . . . . . . . . . . . . 16 ⊢ (p ∈ NC → (1c ≤c p ∨ p ≤c 1c))
61	60	adantl 452	. . . . . . . . . . . . . . 15 ⊢ ((m ∈ NC ∧ p ∈ NC ) → (1c ≤c p ∨ p ≤c 1c))
62		simpll 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ 1c ≤c p) → m ∈ NC )
63	52	a1i 10	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ 1c ≤c p) → 1c ∈ NC )
64		simplr 731	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ 1c ≤c p) → p ∈ NC )
65		simpr 447	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ 1c ≤c p) → 1c ≤c p)
66		leaddc2 6215	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ 1c ∈ NC ∧ p ∈ NC ) ∧ 1c ≤c p) → (m +c 1c) ≤c (m +c p))
67	62, 63, 64, 65, 66	syl31anc 1185	. . . . . . . . . . . . . . . . 17 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ 1c ≤c p) → (m +c 1c) ≤c (m +c p))
68	67	ex 423	. . . . . . . . . . . . . . . 16 ⊢ ((m ∈ NC ∧ p ∈ NC ) → (1c ≤c p → (m +c 1c) ≤c (m +c p)))
69		simpll 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ p ≤c 1c) → m ∈ NC )
70		simplr 731	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ p ≤c 1c) → p ∈ NC )
71	52	a1i 10	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ p ≤c 1c) → 1c ∈ NC )
72		simpr 447	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ p ≤c 1c) → p ≤c 1c)
73		leaddc2 6215	. . . . . . . . . . . . . . . . . 18 ⊢ (((m ∈ NC ∧ p ∈ NC ∧ 1c ∈ NC ) ∧ p ≤c 1c) → (m +c p) ≤c (m +c 1c))
74	69, 70, 71, 72, 73	syl31anc 1185	. . . . . . . . . . . . . . . . 17 ⊢ (((m ∈ NC ∧ p ∈ NC ) ∧ p ≤c 1c) → (m +c p) ≤c (m +c 1c))
75	74	ex 423	. . . . . . . . . . . . . . . 16 ⊢ ((m ∈ NC ∧ p ∈ NC ) → (p ≤c 1c → (m +c p) ≤c (m +c 1c)))
76	68, 75	orim12d 811	. . . . . . . . . . . . . . 15 ⊢ ((m ∈ NC ∧ p ∈ NC ) → ((1c ≤c p ∨ p ≤c 1c) → ((m +c 1c) ≤c (m +c p) ∨ (m +c p) ≤c (m +c 1c))))
77	61, 76	mpd 14	. . . . . . . . . . . . . 14 ⊢ ((m ∈ NC ∧ p ∈ NC ) → ((m +c 1c) ≤c (m +c p) ∨ (m +c p) ≤c (m +c 1c)))
78		breq2 4643	. . . . . . . . . . . . . . . . 17 ⊢ (n = (m +c p) → ((m +c 1c) ≤c n ↔ (m +c 1c) ≤c (m +c p)))
79		breq1 4642	. . . . . . . . . . . . . . . . 17 ⊢ (n = (m +c p) → (n ≤c (m +c 1c) ↔ (m +c p) ≤c (m +c 1c)))
80	78, 79	orbi12d 690	. . . . . . . . . . . . . . . 16 ⊢ (n = (m +c p) → (((m +c 1c) ≤c n ∨ n ≤c (m +c 1c)) ↔ ((m +c 1c) ≤c (m +c p) ∨ (m +c p) ≤c (m +c 1c))))
81	80	biimprd 214	. . . . . . . . . . . . . . 15 ⊢ (n = (m +c p) → (((m +c 1c) ≤c (m +c p) ∨ (m +c p) ≤c (m +c 1c)) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
82	81	com12 27	. . . . . . . . . . . . . 14 ⊢ (((m +c 1c) ≤c (m +c p) ∨ (m +c p) ≤c (m +c 1c)) → (n = (m +c p) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
83	77, 82	syl 15	. . . . . . . . . . . . 13 ⊢ ((m ∈ NC ∧ p ∈ NC ) → (n = (m +c p) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
84	83	rexlimdva 2738	. . . . . . . . . . . 12 ⊢ (m ∈ NC → (∃p ∈ NC n = (m +c p) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
85	84	adantr 451	. . . . . . . . . . 11 ⊢ ((m ∈ NC ∧ n ∈ NC ) → (∃p ∈ NC n = (m +c p) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
86	50, 85	sylbid 206	. . . . . . . . . 10 ⊢ ((m ∈ NC ∧ n ∈ NC ) → (m ≤c n → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
87		addlecncs 6209	. . . . . . . . . . . . . . 15 ⊢ ((m ∈ NC ∧ 1c ∈ NC ) → m ≤c (m +c 1c))
88	52, 87	mpan2 652	. . . . . . . . . . . . . 14 ⊢ (m ∈ NC → m ≤c (m +c 1c))
89	88	adantl 452	. . . . . . . . . . . . 13 ⊢ ((n ∈ NC ∧ m ∈ NC ) → m ≤c (m +c 1c))
90		peano2nc 6145	. . . . . . . . . . . . . . 15 ⊢ (m ∈ NC → (m +c 1c) ∈ NC )
91	90	adantl 452	. . . . . . . . . . . . . 14 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (m +c 1c) ∈ NC )
92		lectr 6211	. . . . . . . . . . . . . 14 ⊢ ((n ∈ NC ∧ m ∈ NC ∧ (m +c 1c) ∈ NC ) → ((n ≤c m ∧ m ≤c (m +c 1c)) → n ≤c (m +c 1c)))
93	91, 92	mpd3an3 1278	. . . . . . . . . . . . 13 ⊢ ((n ∈ NC ∧ m ∈ NC ) → ((n ≤c m ∧ m ≤c (m +c 1c)) → n ≤c (m +c 1c)))
94	89, 93	mpan2d 655	. . . . . . . . . . . 12 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (n ≤c m → n ≤c (m +c 1c)))
95	94	ancoms 439	. . . . . . . . . . 11 ⊢ ((m ∈ NC ∧ n ∈ NC ) → (n ≤c m → n ≤c (m +c 1c)))
96		olc 373	. . . . . . . . . . 11 ⊢ (n ≤c (m +c 1c) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c)))
97	95, 96	syl6 29	. . . . . . . . . 10 ⊢ ((m ∈ NC ∧ n ∈ NC ) → (n ≤c m → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
98	86, 97	jaod 369	. . . . . . . . 9 ⊢ ((m ∈ NC ∧ n ∈ NC ) → ((m ≤c n ∨ n ≤c m) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
99	49, 44, 98	syl2an 463	. . . . . . . 8 ⊢ ((m ∈ Nn ∧ n ∈ Nn ) → ((m ≤c n ∨ n ≤c m) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c))))
100	99	ex 423	. . . . . . 7 ⊢ (m ∈ Nn → (n ∈ Nn → ((m ≤c n ∨ n ≤c m) → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c)))))
101	100	a2d 23	. . . . . 6 ⊢ (m ∈ Nn → ((n ∈ Nn → (m ≤c n ∨ n ≤c m)) → (n ∈ Nn → ((m +c 1c) ≤c n ∨ n ≤c (m +c 1c)))))
102	27, 31, 35, 39, 43, 48, 101	finds 4411	. . . . 5 ⊢ (A ∈ Nn → (n ∈ Nn → (A ≤c n ∨ n ≤c A)))
103	102	com12 27	. . . 4 ⊢ (n ∈ Nn → (A ∈ Nn → (A ≤c n ∨ n ≤c A)))
104	4, 103	vtoclga 2920	. . 3 ⊢ (B ∈ Nn → (A ∈ Nn → (A ≤c B ∨ B ≤c A)))
105	104	com12 27	. 2 ⊢ (A ∈ Nn → (B ∈ Nn → (A ≤c B ∨ B ≤c A)))
106	105	imp 418	1 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (A ≤c B ∨ B ≤c A))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ∪ cun 3207  {csn 3737  1_cc1c 4134   Nn cnnc 4373  0_cc0c 4374   +_c cplc 4375  ⟨cop 4561   class class class wbr 4639   “ cima 4722  ^◡ccnv 4771   NC cncs 6088   ≤_c clec 6089

This theorem was proved from 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 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-fv 4795  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: (None)