Theorem hashtpg 13267

Description: The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.)

Assertion

Ref	Expression
hashtpg	|- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) <-> ( # ` { A , B , C } ) = 3 ) )

Proof of Theorem hashtpg

Step	Hyp	Ref	Expression
1		simpl3 1066	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> C e. W )
2		prfi 8235	. . . . . . 7 |- { A , B } e. Fin
3	2	a1i 11	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> { A , B } e. Fin )
4		elprg 4196	. . . . . . . . . . . . . . . 16 |- ( C e. W -> ( C e. { A , B } <-> ( C = A \/ C = B ) ) )
5		orcom 402	. . . . . . . . . . . . . . . . 17 |- ( ( C = A \/ C = B ) <-> ( C = B \/ C = A ) )
6		nne 2798	. . . . . . . . . . . . . . . . . . 19 |- ( -. B =/= C <-> B = C )
7		eqcom 2629	. . . . . . . . . . . . . . . . . . 19 |- ( B = C <-> C = B )
8	6, 7	bitr2i 265	. . . . . . . . . . . . . . . . . 18 |- ( C = B <-> -. B =/= C )
9		nne 2798	. . . . . . . . . . . . . . . . . . 19 |- ( -. C =/= A <-> C = A )
10	9	bicomi 214	. . . . . . . . . . . . . . . . . 18 |- ( C = A <-> -. C =/= A )
11	8, 10	orbi12i 543	. . . . . . . . . . . . . . . . 17 |- ( ( C = B \/ C = A ) <-> ( -. B =/= C \/ -. C =/= A ) )
12	5, 11	bitri 264	. . . . . . . . . . . . . . . 16 |- ( ( C = A \/ C = B ) <-> ( -. B =/= C \/ -. C =/= A ) )
13	4, 12	syl6bb 276	. . . . . . . . . . . . . . 15 |- ( C e. W -> ( C e. { A , B } <-> ( -. B =/= C \/ -. C =/= A ) ) )
14	13	biimpd 219	. . . . . . . . . . . . . 14 |- ( C e. W -> ( C e. { A , B } -> ( -. B =/= C \/ -. C =/= A ) ) )
15	14	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( C e. { A , B } -> ( -. B =/= C \/ -. C =/= A ) ) )
16	15	imp 445	. . . . . . . . . . . 12 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ C e. { A , B } ) -> ( -. B =/= C \/ -. C =/= A ) )
17	16	olcd 408	. . . . . . . . . . 11 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ C e. { A , B } ) -> ( -. A =/= B \/ ( -. B =/= C \/ -. C =/= A ) ) )
18	17	ex 450	. . . . . . . . . 10 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( C e. { A , B } -> ( -. A =/= B \/ ( -. B =/= C \/ -. C =/= A ) ) ) )
19		3orass 1040	. . . . . . . . . 10 |- ( ( -. A =/= B \/ -. B =/= C \/ -. C =/= A ) <-> ( -. A =/= B \/ ( -. B =/= C \/ -. C =/= A ) ) )
20	18, 19	syl6ibr 242	. . . . . . . . 9 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( C e. { A , B } -> ( -. A =/= B \/ -. B =/= C \/ -. C =/= A ) ) )
21		3ianor 1055	. . . . . . . . 9 |- ( -. ( A =/= B /\ B =/= C /\ C =/= A ) <-> ( -. A =/= B \/ -. B =/= C \/ -. C =/= A ) )
22	20, 21	syl6ibr 242	. . . . . . . 8 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( C e. { A , B } -> -. ( A =/= B /\ B =/= C /\ C =/= A ) ) )
23	22	con2d 129	. . . . . . 7 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) -> -. C e. { A , B } ) )
24	23	imp 445	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> -. C e. { A , B } )
25		hashunsng 13181	. . . . . . 7 |- ( C e. W -> ( ( { A , B } e. Fin /\ -. C e. { A , B } ) -> ( # ` ( { A , B } u. { C } ) ) = ( ( # ` { A , B } ) + 1 ) ) )
26	25	imp 445	. . . . . 6 |- ( ( C e. W /\ ( { A , B } e. Fin /\ -. C e. { A , B } ) ) -> ( # ` ( { A , B } u. { C } ) ) = ( ( # ` { A , B } ) + 1 ) )
27	1, 3, 24, 26	syl12anc 1324	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( # ` ( { A , B } u. { C } ) ) = ( ( # ` { A , B } ) + 1 ) )
28		simpr1 1067	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> A =/= B )
29		3simpa 1058	. . . . . . . . 9 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( A e. U /\ B e. V ) )
30	29	adantr 481	. . . . . . . 8 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( A e. U /\ B e. V ) )
31		hashprg 13182	. . . . . . . 8 |- ( ( A e. U /\ B e. V ) -> ( A =/= B <-> ( # ` { A , B } ) = 2 ) )
32	30, 31	syl 17	. . . . . . 7 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( A =/= B <-> ( # ` { A , B } ) = 2 ) )
33	28, 32	mpbid 222	. . . . . 6 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( # ` { A , B } ) = 2 )
34	33	oveq1d 6665	. . . . 5 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( ( # ` { A , B } ) + 1 ) = ( 2 + 1 ) )
35	27, 34	eqtrd 2656	. . . 4 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( # ` ( { A , B } u. { C } ) ) = ( 2 + 1 ) )
36		df-tp 4182	. . . . 5 |- { A , B , C } = ( { A , B } u. { C } )
37	36	fveq2i 6194	. . . 4 |- ( # ` { A , B , C } ) = ( # ` ( { A , B } u. { C } ) )
38		df-3 11080	. . . 4 |- 3 = ( 2 + 1 )
39	35, 37, 38	3eqtr4g 2681	. . 3 |- ( ( ( A e. U /\ B e. V /\ C e. W ) /\ ( A =/= B /\ B =/= C /\ C =/= A ) ) -> ( # ` { A , B , C } ) = 3 )
40	39	ex 450	. 2 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) -> ( # ` { A , B , C } ) = 3 ) )
41		nne 2798	. . . . . . 7 |- ( -. A =/= B <-> A = B )
42		hashprlei 13250	. . . . . . . . 9 |- ( { B , C } e. Fin /\ ( # ` { B , C } ) <_ 2 )
43		prfi 8235	. . . . . . . . . . . . . 14 |- { B , C } e. Fin
44		hashcl 13147	. . . . . . . . . . . . . . 15 |- ( { B , C } e. Fin -> ( # ` { B , C } ) e. NN0 )
45	44	nn0zd 11480	. . . . . . . . . . . . . 14 |- ( { B , C } e. Fin -> ( # ` { B , C } ) e. ZZ )
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 |- ( # ` { B , C } ) e. ZZ
47		2z 11409	. . . . . . . . . . . . 13 |- 2 e. ZZ
48		zleltp1 11428	. . . . . . . . . . . . . 14 |- ( ( ( # ` { B , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { B , C } ) <_ 2 <-> ( # ` { B , C } ) < ( 2 + 1 ) ) )
49		2p1e3 11151	. . . . . . . . . . . . . . . . 17 |- ( 2 + 1 ) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` { B , C } ) e. ZZ /\ 2 e. ZZ ) -> ( 2 + 1 ) = 3 )
51	50	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( ( ( # ` { B , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { B , C } ) < ( 2 + 1 ) <-> ( # ` { B , C } ) < 3 ) )
52	51	biimpd 219	. . . . . . . . . . . . . 14 |- ( ( ( # ` { B , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { B , C } ) < ( 2 + 1 ) -> ( # ` { B , C } ) < 3 ) )
53	48, 52	sylbid 230	. . . . . . . . . . . . 13 |- ( ( ( # ` { B , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { B , C } ) <_ 2 -> ( # ` { B , C } ) < 3 ) )
54	46, 47, 53	mp2an 708	. . . . . . . . . . . 12 |- ( ( # ` { B , C } ) <_ 2 -> ( # ` { B , C } ) < 3 )
55	44	nn0red 11352	. . . . . . . . . . . . . 14 |- ( { B , C } e. Fin -> ( # ` { B , C } ) e. RR )
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 |- ( # ` { B , C } ) e. RR
57		3re 11094	. . . . . . . . . . . . 13 |- 3 e. RR
58	56, 57	ltnei 10161	. . . . . . . . . . . 12 |- ( ( # ` { B , C } ) < 3 -> 3 =/= ( # ` { B , C } ) )
59	54, 58	syl 17	. . . . . . . . . . 11 |- ( ( # ` { B , C } ) <_ 2 -> 3 =/= ( # ` { B , C } ) )
60	59	necomd 2849	. . . . . . . . . 10 |- ( ( # ` { B , C } ) <_ 2 -> ( # ` { B , C } ) =/= 3 )
61	60	adantl 482	. . . . . . . . 9 |- ( ( { B , C } e. Fin /\ ( # ` { B , C } ) <_ 2 ) -> ( # ` { B , C } ) =/= 3 )
62	42, 61	mp1i 13	. . . . . . . 8 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { B , C } ) =/= 3 )
63		tpeq1 4277	. . . . . . . . . . 11 |- ( A = B -> { A , B , C } = { B , B , C } )
64		tpidm12 4290	. . . . . . . . . . 11 |- { B , B , C } = { B , C }
65	63, 64	syl6req 2673	. . . . . . . . . 10 |- ( A = B -> { B , C } = { A , B , C } )
66	65	fveq2d 6195	. . . . . . . . 9 |- ( A = B -> ( # ` { B , C } ) = ( # ` { A , B , C } ) )
67	66	neeq1d 2853	. . . . . . . 8 |- ( A = B -> ( ( # ` { B , C } ) =/= 3 <-> ( # ` { A , B , C } ) =/= 3 ) )
68	62, 67	syl5ib 234	. . . . . . 7 |- ( A = B -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
69	41, 68	sylbi 207	. . . . . 6 |- ( -. A =/= B -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
70		hashprlei 13250	. . . . . . . . 9 |- ( { A , C } e. Fin /\ ( # ` { A , C } ) <_ 2 )
71		prfi 8235	. . . . . . . . . . . . . 14 |- { A , C } e. Fin
72		hashcl 13147	. . . . . . . . . . . . . . 15 |- ( { A , C } e. Fin -> ( # ` { A , C } ) e. NN0 )
73	72	nn0zd 11480	. . . . . . . . . . . . . 14 |- ( { A , C } e. Fin -> ( # ` { A , C } ) e. ZZ )
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 |- ( # ` { A , C } ) e. ZZ
75		zleltp1 11428	. . . . . . . . . . . . . 14 |- ( ( ( # ` { A , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , C } ) <_ 2 <-> ( # ` { A , C } ) < ( 2 + 1 ) ) )
76	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` { A , C } ) e. ZZ /\ 2 e. ZZ ) -> ( 2 + 1 ) = 3 )
77	76	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( ( ( # ` { A , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , C } ) < ( 2 + 1 ) <-> ( # ` { A , C } ) < 3 ) )
78	77	biimpd 219	. . . . . . . . . . . . . 14 |- ( ( ( # ` { A , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , C } ) < ( 2 + 1 ) -> ( # ` { A , C } ) < 3 ) )
79	75, 78	sylbid 230	. . . . . . . . . . . . 13 |- ( ( ( # ` { A , C } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , C } ) <_ 2 -> ( # ` { A , C } ) < 3 ) )
80	74, 47, 79	mp2an 708	. . . . . . . . . . . 12 |- ( ( # ` { A , C } ) <_ 2 -> ( # ` { A , C } ) < 3 )
81	72	nn0red 11352	. . . . . . . . . . . . . 14 |- ( { A , C } e. Fin -> ( # ` { A , C } ) e. RR )
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 |- ( # ` { A , C } ) e. RR
83	82, 57	ltnei 10161	. . . . . . . . . . . 12 |- ( ( # ` { A , C } ) < 3 -> 3 =/= ( # ` { A , C } ) )
84	80, 83	syl 17	. . . . . . . . . . 11 |- ( ( # ` { A , C } ) <_ 2 -> 3 =/= ( # ` { A , C } ) )
85	84	necomd 2849	. . . . . . . . . 10 |- ( ( # ` { A , C } ) <_ 2 -> ( # ` { A , C } ) =/= 3 )
86	85	adantl 482	. . . . . . . . 9 |- ( ( { A , C } e. Fin /\ ( # ` { A , C } ) <_ 2 ) -> ( # ` { A , C } ) =/= 3 )
87	70, 86	mp1i 13	. . . . . . . 8 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , C } ) =/= 3 )
88		tpeq2 4278	. . . . . . . . . . 11 |- ( B = C -> { A , B , C } = { A , C , C } )
89		tpidm23 4292	. . . . . . . . . . 11 |- { A , C , C } = { A , C }
90	88, 89	syl6req 2673	. . . . . . . . . 10 |- ( B = C -> { A , C } = { A , B , C } )
91	90	fveq2d 6195	. . . . . . . . 9 |- ( B = C -> ( # ` { A , C } ) = ( # ` { A , B , C } ) )
92	91	neeq1d 2853	. . . . . . . 8 |- ( B = C -> ( ( # ` { A , C } ) =/= 3 <-> ( # ` { A , B , C } ) =/= 3 ) )
93	87, 92	syl5ib 234	. . . . . . 7 |- ( B = C -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
94	6, 93	sylbi 207	. . . . . 6 |- ( -. B =/= C -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
95		hashprlei 13250	. . . . . . . . 9 |- ( { A , B } e. Fin /\ ( # ` { A , B } ) <_ 2 )
96		hashcl 13147	. . . . . . . . . . . . . . 15 |- ( { A , B } e. Fin -> ( # ` { A , B } ) e. NN0 )
97	96	nn0zd 11480	. . . . . . . . . . . . . 14 |- ( { A , B } e. Fin -> ( # ` { A , B } ) e. ZZ )
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 |- ( # ` { A , B } ) e. ZZ
99		zleltp1 11428	. . . . . . . . . . . . . 14 |- ( ( ( # ` { A , B } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , B } ) <_ 2 <-> ( # ` { A , B } ) < ( 2 + 1 ) ) )
100	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` { A , B } ) e. ZZ /\ 2 e. ZZ ) -> ( 2 + 1 ) = 3 )
101	100	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( ( ( # ` { A , B } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , B } ) < ( 2 + 1 ) <-> ( # ` { A , B } ) < 3 ) )
102	101	biimpd 219	. . . . . . . . . . . . . 14 |- ( ( ( # ` { A , B } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , B } ) < ( 2 + 1 ) -> ( # ` { A , B } ) < 3 ) )
103	99, 102	sylbid 230	. . . . . . . . . . . . 13 |- ( ( ( # ` { A , B } ) e. ZZ /\ 2 e. ZZ ) -> ( ( # ` { A , B } ) <_ 2 -> ( # ` { A , B } ) < 3 ) )
104	98, 47, 103	mp2an 708	. . . . . . . . . . . 12 |- ( ( # ` { A , B } ) <_ 2 -> ( # ` { A , B } ) < 3 )
105	96	nn0red 11352	. . . . . . . . . . . . . 14 |- ( { A , B } e. Fin -> ( # ` { A , B } ) e. RR )
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 |- ( # ` { A , B } ) e. RR
107	106, 57	ltnei 10161	. . . . . . . . . . . 12 |- ( ( # ` { A , B } ) < 3 -> 3 =/= ( # ` { A , B } ) )
108	104, 107	syl 17	. . . . . . . . . . 11 |- ( ( # ` { A , B } ) <_ 2 -> 3 =/= ( # ` { A , B } ) )
109	108	necomd 2849	. . . . . . . . . 10 |- ( ( # ` { A , B } ) <_ 2 -> ( # ` { A , B } ) =/= 3 )
110	109	adantl 482	. . . . . . . . 9 |- ( ( { A , B } e. Fin /\ ( # ` { A , B } ) <_ 2 ) -> ( # ` { A , B } ) =/= 3 )
111	95, 110	mp1i 13	. . . . . . . 8 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B } ) =/= 3 )
112		tpeq3 4279	. . . . . . . . . . 11 |- ( C = A -> { A , B , C } = { A , B , A } )
113		tpidm13 4291	. . . . . . . . . . 11 |- { A , B , A } = { A , B }
114	112, 113	syl6req 2673	. . . . . . . . . 10 |- ( C = A -> { A , B } = { A , B , C } )
115	114	fveq2d 6195	. . . . . . . . 9 |- ( C = A -> ( # ` { A , B } ) = ( # ` { A , B , C } ) )
116	115	neeq1d 2853	. . . . . . . 8 |- ( C = A -> ( ( # ` { A , B } ) =/= 3 <-> ( # ` { A , B , C } ) =/= 3 ) )
117	111, 116	syl5ib 234	. . . . . . 7 |- ( C = A -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
118	9, 117	sylbi 207	. . . . . 6 |- ( -. C =/= A -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
119	69, 94, 118	3jaoi 1391	. . . . 5 |- ( ( -. A =/= B \/ -. B =/= C \/ -. C =/= A ) -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
120	21, 119	sylbi 207	. . . 4 |- ( -. ( A =/= B /\ B =/= C /\ C =/= A ) -> ( ( A e. U /\ B e. V /\ C e. W ) -> ( # ` { A , B , C } ) =/= 3 ) )
121	120	com12 32	. . 3 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( -. ( A =/= B /\ B =/= C /\ C =/= A ) -> ( # ` { A , B , C } ) =/= 3 ) )
122	121	necon4bd 2814	. 2 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( # ` { A , B , C } ) = 3 -> ( A =/= B /\ B =/= C /\ C =/= A ) ) )
123	40, 122	impbid 202	1 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) <-> ( # ` { A , B , C } ) = 3 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   2c2 11070   3c3 11071   ZZcz 11377   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  hashge3el3dif  13268  konigsberglem5  27118  poimirlem9  33418