Theorem nndisjeq 4430

Description: Either two naturals are disjoint or they are the same natural. Theorem X.1.18 of [Rosser] p. 526. (Contributed by SF, 17-Jan-2015.)

Assertion

Ref	Expression
nndisjeq	|- ((M e. Nn /\ N e. Nn ) -> ((M i^i N) = (/) \/ M = N))

Proof of Theorem nndisjeq

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

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . . 8 |- p e. _V
2	1	elcompl 3226	. . . . . . 7 |- (p e. ∼ ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ) <-> -. p e. ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ))
3	1	elimak 4260	. . . . . . . . 9 |- (p e. ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ) <-> E.n e. Nn <<n, p>> e. ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ))
4		opkex 4114	. . . . . . . . . . . 12 |- <<n, p>> e. _V
5	4	elcompl 3226	. . . . . . . . . . 11 |- (<<n, p>> e. ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ) <-> -. <<n, p>> e. ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ))
6		elun 3221	. . . . . . . . . . . 12 |- (<<n, p>> e. ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ) <-> (<<n, p>> e. ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \/ <<n, p>> e. _I_kk ))
7		vex 2863	. . . . . . . . . . . . . . . 16 |- n e. _V
8	7, 1	ndisjrelk 4324	. . . . . . . . . . . . . . 15 |- (<<n, p>> e. (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> (n i^i p) =/= (/))
9	8	notbii 287	. . . . . . . . . . . . . 14 |- (-. <<n, p>> e. (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> -. (n i^i p) =/= (/))
10	4	elcompl 3226	. . . . . . . . . . . . . 14 |- (<<n, p>> e. ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> -. <<n, p>> e. (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c))
11		df-ne 2519	. . . . . . . . . . . . . . 15 |- ((n i^i p) =/= (/) <-> -. (n i^i p) = (/))
12	11	con2bii 322	. . . . . . . . . . . . . 14 |- ((n i^i p) = (/) <-> -. (n i^i p) =/= (/))
13	9, 10, 12	3bitr4i 268	. . . . . . . . . . . . 13 |- (<<n, p>> e. ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> (n i^i p) = (/))
14		opkelidkg 4275	. . . . . . . . . . . . . 14 |- ((n e. _V /\ p e. _V) -> (<<n, p>> e. _I_kk <-> n = p))
15	7, 1, 14	mp2an 653	. . . . . . . . . . . . 13 |- (<<n, p>> e. _I_kk <-> n = p)
16	13, 15	orbi12i 507	. . . . . . . . . . . 12 |- ((<<n, p>> e. ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \/ <<n, p>> e. _I_kk ) <-> ((n i^i p) = (/) \/ n = p))
17		incom 3449	. . . . . . . . . . . . . 14 |- (n i^i p) = (p i^i n)
18	17	eqeq1i 2360	. . . . . . . . . . . . 13 |- ((n i^i p) = (/) <-> (p i^i n) = (/))
19		eqcom 2355	. . . . . . . . . . . . 13 |- (n = p <-> p = n)
20	18, 19	orbi12i 507	. . . . . . . . . . . 12 |- (((n i^i p) = (/) \/ n = p) <-> ((p i^i n) = (/) \/ p = n))
21	6, 16, 20	3bitri 262	. . . . . . . . . . 11 |- (<<n, p>> e. ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ) <-> ((p i^i n) = (/) \/ p = n))
22	5, 21	xchbinx 301	. . . . . . . . . 10 |- (<<n, p>> e. ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ) <-> -. ((p i^i n) = (/) \/ p = n))
23	22	rexbii 2640	. . . . . . . . 9 |- (E.n e. Nn <<n, p>> e. ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ) <-> E.n e. Nn -. ((p i^i n) = (/) \/ p = n))
24		rexnal 2626	. . . . . . . . 9 |- (E.n e. Nn -. ((p i^i n) = (/) \/ p = n) <-> -. A.n e. Nn ((p i^i n) = (/) \/ p = n))
25	3, 23, 24	3bitri 262	. . . . . . . 8 |- (p e. ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ) <-> -. A.n e. Nn ((p i^i n) = (/) \/ p = n))
26	25	con2bii 322	. . . . . . 7 |- (A.n e. Nn ((p i^i n) = (/) \/ p = n) <-> -. p e. ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ))
27	2, 26	bitr4i 243	. . . . . 6 |- (p e. ∼ ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ) <-> A.n e. Nn ((p i^i n) = (/) \/ p = n))
28	27	eqabi 2465	. . . . 5 |- ∼ ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ) = {p | A.n e. Nn ((p i^i n) = (/) \/ p = n)}
29		ssetkex 4295	. . . . . . . . . . . . 13 |- Sk e. _V
30	29	ins3kex 4309	. . . . . . . . . . . 12 |- Ins3k Sk e. _V
31	29	ins2kex 4308	. . . . . . . . . . . 12 |- Ins2k Sk e. _V
32	30, 31	inex 4106	. . . . . . . . . . 11 |- ( Ins3k Sk i^i Ins2k Sk ) e. _V
33		1cex 4143	. . . . . . . . . . . . 13 |- 1c e. _V
34	33	pw1ex 4304	. . . . . . . . . . . 12 |- ~P1 1c e. _V
35	34	pw1ex 4304	. . . . . . . . . . 11 |- ~P1 ~P1 1c e. _V
36	32, 35	imakex 4301	. . . . . . . . . 10 |- (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) e. _V
37	36	complex 4105	. . . . . . . . 9 |- ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) e. _V
38		idkex 4315	. . . . . . . . 9 |- _I_kk e. _V
39	37, 38	unex 4107	. . . . . . . 8 |- ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ) e. _V
40	39	complex 4105	. . . . . . 7 |- ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk ) e. _V
41		nncex 4397	. . . . . . 7 |- Nn e. _V
42	40, 41	imakex 4301	. . . . . 6 |- ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ) e. _V
43	42	complex 4105	. . . . 5 |- ∼ ( ∼ ( ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) u. _I_kk )"k Nn ) e. _V
44	28, 43	eqeltrri 2424	. . . 4 |- {p | A.n e. Nn ((p i^i n) = (/) \/ p = n)} e. _V
45		df-0c 4378	. . . . . . . . . . 11 |- 0c = {(/)}
46	45	eqeq2i 2363	. . . . . . . . . 10 |- (p = 0c <-> p = {(/)})
47	46	biimpi 186	. . . . . . . . 9 |- (p = 0c -> p = {(/)})
48	47	ineq1d 3457	. . . . . . . 8 |- (p = 0c -> (p i^i n) = ({(/)} i^i n))
49	48	eqeq1d 2361	. . . . . . 7 |- (p = 0c -> ((p i^i n) = (/) <-> ({(/)} i^i n) = (/)))
50		incom 3449	. . . . . . . . 9 |- ({(/)} i^i n) = (n i^i {(/)})
51	50	eqeq1i 2360	. . . . . . . 8 |- (({(/)} i^i n) = (/) <-> (n i^i {(/)}) = (/))
52		disjsn 3787	. . . . . . . 8 |- ((n i^i {(/)}) = (/) <-> -. (/) e. n)
53	51, 52	bitri 240	. . . . . . 7 |- (({(/)} i^i n) = (/) <-> -. (/) e. n)
54	49, 53	syl6bb 252	. . . . . 6 |- (p = 0c -> ((p i^i n) = (/) <-> -. (/) e. n))
55		eqeq1 2359	. . . . . . 7 |- (p = 0c -> (p = n <-> 0c = n))
56		eqcom 2355	. . . . . . 7 |- (0c = n <-> n = 0c)
57	55, 56	syl6bb 252	. . . . . 6 |- (p = 0c -> (p = n <-> n = 0c))
58	54, 57	orbi12d 690	. . . . 5 |- (p = 0c -> (((p i^i n) = (/) \/ p = n) <-> (-. (/) e. n \/ n = 0c)))
59	58	ralbidv 2635	. . . 4 |- (p = 0c -> (A.n e. Nn ((p i^i n) = (/) \/ p = n) <-> A.n e. Nn (-. (/) e. n \/ n = 0c)))
60		ineq1 3451	. . . . . . . 8 |- (p = m -> (p i^i n) = (m i^i n))
61	60	eqeq1d 2361	. . . . . . 7 |- (p = m -> ((p i^i n) = (/) <-> (m i^i n) = (/)))
62		eqeq1 2359	. . . . . . 7 |- (p = m -> (p = n <-> m = n))
63	61, 62	orbi12d 690	. . . . . 6 |- (p = m -> (((p i^i n) = (/) \/ p = n) <-> ((m i^i n) = (/) \/ m = n)))
64	63	ralbidv 2635	. . . . 5 |- (p = m -> (A.n e. Nn ((p i^i n) = (/) \/ p = n) <-> A.n e. Nn ((m i^i n) = (/) \/ m = n)))
65		ineq2 3452	. . . . . . . 8 |- (n = q -> (m i^i n) = (m i^i q))
66	65	eqeq1d 2361	. . . . . . 7 |- (n = q -> ((m i^i n) = (/) <-> (m i^i q) = (/)))
67		equequ2 1686	. . . . . . 7 |- (n = q -> (m = n <-> m = q))
68	66, 67	orbi12d 690	. . . . . 6 |- (n = q -> (((m i^i n) = (/) \/ m = n) <-> ((m i^i q) = (/) \/ m = q)))
69	68	cbvralv 2836	. . . . 5 |- (A.n e. Nn ((m i^i n) = (/) \/ m = n) <-> A.q e. Nn ((m i^i q) = (/) \/ m = q))
70	64, 69	syl6bb 252	. . . 4 |- (p = m -> (A.n e. Nn ((p i^i n) = (/) \/ p = n) <-> A.q e. Nn ((m i^i q) = (/) \/ m = q)))
71		ineq1 3451	. . . . . . 7 |- (p = (m +o 1c) -> (p i^i n) = ((m +o 1c) i^i n))
72	71	eqeq1d 2361	. . . . . 6 |- (p = (m +o 1c) -> ((p i^i n) = (/) <-> ((m +o 1c) i^i n) = (/)))
73		eqeq1 2359	. . . . . 6 |- (p = (m +o 1c) -> (p = n <-> (m +o 1c) = n))
74	72, 73	orbi12d 690	. . . . 5 |- (p = (m +o 1c) -> (((p i^i n) = (/) \/ p = n) <-> (((m +o 1c) i^i n) = (/) \/ (m +o 1c) = n)))
75	74	ralbidv 2635	. . . 4 |- (p = (m +o 1c) -> (A.n e. Nn ((p i^i n) = (/) \/ p = n) <-> A.n e. Nn (((m +o 1c) i^i n) = (/) \/ (m +o 1c) = n)))
76		ineq1 3451	. . . . . . 7 |- (p = M -> (p i^i n) = (M i^i n))
77	76	eqeq1d 2361	. . . . . 6 |- (p = M -> ((p i^i n) = (/) <-> (M i^i n) = (/)))
78		eqeq1 2359	. . . . . 6 |- (p = M -> (p = n <-> M = n))
79	77, 78	orbi12d 690	. . . . 5 |- (p = M -> (((p i^i n) = (/) \/ p = n) <-> ((M i^i n) = (/) \/ M = n)))
80	79	ralbidv 2635	. . . 4 |- (p = M -> (A.n e. Nn ((p i^i n) = (/) \/ p = n) <-> A.n e. Nn ((M i^i n) = (/) \/ M = n)))
81		nnc0suc 4413	. . . . . . 7 |- (n e. Nn <-> (n = 0c \/ E.m e. Nn n = (m +o 1c)))
82		0nelsuc 4401	. . . . . . . . . . . 12 |- -. (/) e. (m +o 1c)
83		eleq2 2414	. . . . . . . . . . . . 13 |- (n = (m +o 1c) -> ((/) e. n <-> (/) e. (m +o 1c)))
84	83	biimpcd 215	. . . . . . . . . . . 12 |- ((/) e. n -> (n = (m +o 1c) -> (/) e. (m +o 1c)))
85	82, 84	mtoi 169	. . . . . . . . . . 11 |- ((/) e. n -> -. n = (m +o 1c))
86	85	adantr 451	. . . . . . . . . 10 |- (((/) e. n /\ m e. Nn ) -> -. n = (m +o 1c))
87	86	nrexdv 2718	. . . . . . . . 9 |- ((/) e. n -> -. E.m e. Nn n = (m +o 1c))
88		orel2 372	. . . . . . . . 9 |- (-. E.m e. Nn n = (m +o 1c) -> ((n = 0c \/ E.m e. Nn n = (m +o 1c)) -> n = 0c))
89	87, 88	syl 15	. . . . . . . 8 |- ((/) e. n -> ((n = 0c \/ E.m e. Nn n = (m +o 1c)) -> n = 0c))
90	89	com12 27	. . . . . . 7 |- ((n = 0c \/ E.m e. Nn n = (m +o 1c)) -> ((/) e. n -> n = 0c))
91	81, 90	sylbi 187	. . . . . 6 |- (n e. Nn -> ((/) e. n -> n = 0c))
92		imor 401	. . . . . 6 |- (((/) e. n -> n = 0c) <-> (-. (/) e. n \/ n = 0c))
93	91, 92	sylib 188	. . . . 5 |- (n e. Nn -> (-. (/) e. n \/ n = 0c))
94	93	rgen 2680	. . . 4 |- A.n e. Nn (-. (/) e. n \/ n = 0c)
95		neq0 3561	. . . . . . . . 9 |- (-. ((m +o 1c) i^i n) = (/) <-> E.a a e. ((m +o 1c) i^i n))
96		elin 3220	. . . . . . . . . . 11 |- (a e. ((m +o 1c) i^i n) <-> (a e. (m +o 1c) /\ a e. n))
97		elsuc 4414	. . . . . . . . . . . . 13 |- (a e. (m +o 1c) <-> E.b e. m E.x e. ∼ ba = (b u. {x}))
98		vex 2863	. . . . . . . . . . . . . . . . 17 |- x e. _V
99	98	elcompl 3226	. . . . . . . . . . . . . . . 16 |- (x e. ∼ b <-> -. x e. b)
100	99	anbi2i 675	. . . . . . . . . . . . . . 15 |- ((b e. m /\ x e. ∼ b) <-> (b e. m /\ -. x e. b))
101		simp1r 980	. . . . . . . . . . . . . . . . . . 19 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> n e. Nn )
102		nnc0suc 4413	. . . . . . . . . . . . . . . . . . 19 |- (n e. Nn <-> (n = 0c \/ E.p e. Nn n = (p +o 1c)))
103	101, 102	sylib 188	. . . . . . . . . . . . . . . . . 18 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> (n = 0c \/ E.p e. Nn n = (p +o 1c)))
104		ssun2 3428	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- {x} C_ (b u. {x})
105	98	snid 3761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- x e. {x}
106	104, 105	sselii 3271	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- x e. (b u. {x})
107		n0i 3556	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x e. (b u. {x}) -> -. (b u. {x}) = (/))
108	106, 107	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 |- -. (b u. {x}) = (/)
109	45	eleq2i 2417	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((b u. {x}) e. 0c <-> (b u. {x}) e. {(/)})
110		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- b e. _V
111		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- {x} e. _V
112	110, 111	unex 4107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (b u. {x}) e. _V
113	112	elsnc 3757	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((b u. {x}) e. {(/)} <-> (b u. {x}) = (/))
114	109, 113	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((b u. {x}) e. 0c <-> (b u. {x}) = (/))
115	108, 114	mtbir 290	. . . . . . . . . . . . . . . . . . . . . . 23 |- -. (b u. {x}) e. 0c
116		eleq2 2414	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (n = 0c -> ((b u. {x}) e. n <-> (b u. {x}) e. 0c))
117	116	biimpcd 215	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((b u. {x}) e. n -> (n = 0c -> (b u. {x}) e. 0c))
118	115, 117	mtoi 169	. . . . . . . . . . . . . . . . . . . . . 22 |- ((b u. {x}) e. n -> -. n = 0c)
119	118	adantl 452	. . . . . . . . . . . . . . . . . . . . 21 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) -> -. n = 0c)
120		orel1 371	. . . . . . . . . . . . . . . . . . . . 21 |- (-. n = 0c -> ((n = 0c \/ E.p e. Nn n = (p +o 1c)) -> E.p e. Nn n = (p +o 1c)))
121	119, 120	syl 15	. . . . . . . . . . . . . . . . . . . 20 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) -> ((n = 0c \/ E.p e. Nn n = (p +o 1c)) -> E.p e. Nn n = (p +o 1c)))
122		simpll 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) /\ (b u. {x}) e. (p +o 1c)) -> p e. Nn )
123		simpr3r 1017	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) -> -. x e. b)
124	123	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) /\ (b u. {x}) e. (p +o 1c)) -> -. x e. b)
125		simpr 447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) /\ (b u. {x}) e. (p +o 1c)) -> (b u. {x}) e. (p +o 1c))
126	110, 98	nnsucelr 4429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((p e. Nn /\ (-. x e. b /\ (b u. {x}) e. (p +o 1c))) -> b e. p)
127	122, 124, 125, 126	syl12anc 1180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) /\ (b u. {x}) e. (p +o 1c)) -> b e. p)
128	127	ex 423	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) -> ((b u. {x}) e. (p +o 1c) -> b e. p))
129		ineq2 3452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- (q = p -> (m i^i q) = (m i^i p))
130	129	eqeq1d 2361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (q = p -> ((m i^i q) = (/) <-> (m i^i p) = (/)))
131		equequ2 1686	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (q = p -> (m = q <-> m = p))
132	130, 131	orbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (q = p -> (((m i^i q) = (/) \/ m = q) <-> ((m i^i p) = (/) \/ m = p)))
133	132	rspccv 2953	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (A.q e. Nn ((m i^i q) = (/) \/ m = q) -> (p e. Nn -> ((m i^i p) = (/) \/ m = p)))
134		elin 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- (b e. (m i^i p) <-> (b e. m /\ b e. p))
135		n0i 3556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- (b e. (m i^i p) -> -. (m i^i p) = (/))
136	134, 135	sylbir 204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ((b e. m /\ b e. p) -> -. (m i^i p) = (/))
137		pm2.53 362	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (((m i^i p) = (/) \/ m = p) -> (-. (m i^i p) = (/) -> m = p))
138	136, 137	syl5 28	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (((m i^i p) = (/) \/ m = p) -> ((b e. m /\ b e. p) -> m = p))
139	138	exp3a 425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (((m i^i p) = (/) \/ m = p) -> (b e. m -> (b e. p -> m = p)))
140	133, 139	syl6 29	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (A.q e. Nn ((m i^i q) = (/) \/ m = q) -> (p e. Nn -> (b e. m -> (b e. p -> m = p))))
141	140	com23 72	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (A.q e. Nn ((m i^i q) = (/) \/ m = q) -> (b e. m -> (p e. Nn -> (b e. p -> m = p))))
142	141	imp 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ b e. m) -> (p e. Nn -> (b e. p -> m = p)))
143	142	adantrr 697	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> (p e. Nn -> (b e. p -> m = p)))
144	143	3adant1 973	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> (p e. Nn -> (b e. p -> m = p)))
145	144	impcom 419	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) -> (b e. p -> m = p))
146	128, 145	syld 40	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((p e. Nn /\ ((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b))) -> ((b u. {x}) e. (p +o 1c) -> m = p))
147	146	ex 423	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (p e. Nn -> (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> ((b u. {x}) e. (p +o 1c) -> m = p)))
148	147	com3l 75	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> ((b u. {x}) e. (p +o 1c) -> (p e. Nn -> m = p)))
149	148	imp 418	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. (p +o 1c)) -> (p e. Nn -> m = p))
150		addceq1 4384	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (m = p -> (m +o 1c) = (p +o 1c))
151	149, 150	syl6 29	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. (p +o 1c)) -> (p e. Nn -> (m +o 1c) = (p +o 1c)))
152		eleq2 2414	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (n = (p +o 1c) -> ((b u. {x}) e. n <-> (b u. {x}) e. (p +o 1c)))
153	152	anbi2d 684	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (n = (p +o 1c) -> ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) <-> (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. (p +o 1c))))
154		eqeq2 2362	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (n = (p +o 1c) -> ((m +o 1c) = n <-> (m +o 1c) = (p +o 1c)))
155	154	imbi2d 307	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (n = (p +o 1c) -> ((p e. Nn -> (m +o 1c) = n) <-> (p e. Nn -> (m +o 1c) = (p +o 1c))))
156	153, 155	imbi12d 311	. . . . . . . . . . . . . . . . . . . . . . 23 |- (n = (p +o 1c) -> (((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) -> (p e. Nn -> (m +o 1c) = n)) <-> ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. (p +o 1c)) -> (p e. Nn -> (m +o 1c) = (p +o 1c)))))
157	151, 156	mpbiri 224	. . . . . . . . . . . . . . . . . . . . . 22 |- (n = (p +o 1c) -> ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) -> (p e. Nn -> (m +o 1c) = n)))
158	157	com3l 75	. . . . . . . . . . . . . . . . . . . . 21 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) -> (p e. Nn -> (n = (p +o 1c) -> (m +o 1c) = n)))
159	158	rexlimdv 2738	. . . . . . . . . . . . . . . . . . . 20 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) -> (E.p e. Nn n = (p +o 1c) -> (m +o 1c) = n))
160	121, 159	syld 40	. . . . . . . . . . . . . . . . . . 19 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) /\ (b u. {x}) e. n) -> ((n = 0c \/ E.p e. Nn n = (p +o 1c)) -> (m +o 1c) = n))
161	160	ex 423	. . . . . . . . . . . . . . . . . 18 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> ((b u. {x}) e. n -> ((n = 0c \/ E.p e. Nn n = (p +o 1c)) -> (m +o 1c) = n)))
162	103, 161	mpid 37	. . . . . . . . . . . . . . . . 17 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q) /\ (b e. m /\ -. x e. b)) -> ((b u. {x}) e. n -> (m +o 1c) = n))
163	162	3expa 1151	. . . . . . . . . . . . . . . 16 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) /\ (b e. m /\ -. x e. b)) -> ((b u. {x}) e. n -> (m +o 1c) = n))
164		eleq1 2413	. . . . . . . . . . . . . . . . 17 |- (a = (b u. {x}) -> (a e. n <-> (b u. {x}) e. n))
165	164	imbi1d 308	. . . . . . . . . . . . . . . 16 |- (a = (b u. {x}) -> ((a e. n -> (m +o 1c) = n) <-> ((b u. {x}) e. n -> (m +o 1c) = n)))
166	163, 165	syl5ibrcom 213	. . . . . . . . . . . . . . 15 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) /\ (b e. m /\ -. x e. b)) -> (a = (b u. {x}) -> (a e. n -> (m +o 1c) = n)))
167	100, 166	sylan2b 461	. . . . . . . . . . . . . 14 |- ((((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) /\ (b e. m /\ x e. ∼ b)) -> (a = (b u. {x}) -> (a e. n -> (m +o 1c) = n)))
168	167	rexlimdvva 2746	. . . . . . . . . . . . 13 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) -> (E.b e. m E.x e. ∼ ba = (b u. {x}) -> (a e. n -> (m +o 1c) = n)))
169	97, 168	syl5bi 208	. . . . . . . . . . . 12 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) -> (a e. (m +o 1c) -> (a e. n -> (m +o 1c) = n)))
170	169	imp3a 420	. . . . . . . . . . 11 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) -> ((a e. (m +o 1c) /\ a e. n) -> (m +o 1c) = n))
171	96, 170	syl5bi 208	. . . . . . . . . 10 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) -> (a e. ((m +o 1c) i^i n) -> (m +o 1c) = n))
172	171	exlimdv 1636	. . . . . . . . 9 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) -> (E.a a e. ((m +o 1c) i^i n) -> (m +o 1c) = n))
173	95, 172	syl5bi 208	. . . . . . . 8 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) -> (-. ((m +o 1c) i^i n) = (/) -> (m +o 1c) = n))
174	173	orrd 367	. . . . . . 7 |- (((m e. Nn /\ n e. Nn ) /\ A.q e. Nn ((m i^i q) = (/) \/ m = q)) -> (((m +o 1c) i^i n) = (/) \/ (m +o 1c) = n))
175	174	exp31 587	. . . . . 6 |- (m e. Nn -> (n e. Nn -> (A.q e. Nn ((m i^i q) = (/) \/ m = q) -> (((m +o 1c) i^i n) = (/) \/ (m +o 1c) = n))))
176	175	com23 72	. . . . 5 |- (m e. Nn -> (A.q e. Nn ((m i^i q) = (/) \/ m = q) -> (n e. Nn -> (((m +o 1c) i^i n) = (/) \/ (m +o 1c) = n))))
177	176	ralrimdv 2704	. . . 4 |- (m e. Nn -> (A.q e. Nn ((m i^i q) = (/) \/ m = q) -> A.n e. Nn (((m +o 1c) i^i n) = (/) \/ (m +o 1c) = n)))
178	44, 59, 70, 75, 80, 94, 177	finds 4412	. . 3 |- (M e. Nn -> A.n e. Nn ((M i^i n) = (/) \/ M = n))
179		ineq2 3452	. . . . . 6 |- (n = N -> (M i^i n) = (M i^i N))
180	179	eqeq1d 2361	. . . . 5 |- (n = N -> ((M i^i n) = (/) <-> (M i^i N) = (/)))
181		eqeq2 2362	. . . . 5 |- (n = N -> (M = n <-> M = N))
182	180, 181	orbi12d 690	. . . 4 |- (n = N -> (((M i^i n) = (/) \/ M = n) <-> ((M i^i N) = (/) \/ M = N)))
183	182	rspccv 2953	. . 3 |- (A.n e. Nn ((M i^i n) = (/) \/ M = n) -> (N e. Nn -> ((M i^i N) = (/) \/ M = N)))
184	178, 183	syl 15	. 2 |- (M e. Nn -> (N e. Nn -> ((M i^i N) = (/) \/ M = N)))
185	184	imp 418	1 |- ((M e. Nn /\ N e. Nn ) -> ((M i^i N) = (/) \/ M = N))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  A.wral 2615  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   u. cun 3208   i^i cin 3209  (/)c0 3551  {csn 3738  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   S_k cssetk 4184   _I_k_k cidk 4185   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376

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-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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-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-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  nnceleq  4431  sfinltfin  4536