Theorem xposdif 9658

Description: Extended real version of posdif 8210. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)

Assertion

Ref	Expression
xposdif	|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> 0 < ( B +e -e A ) ) )

Proof of Theorem xposdif

Step	Hyp	Ref	Expression
1		elxr 9556	. . 3 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		elxr 9556	. . . . 5 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
3		posdif 8210	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
4		rexsub 9629	. . . . . . . . . 10 |- ( ( B e. RR /\ A e. RR ) -> ( B +e -e A ) = ( B - A ) )
5	4	ancoms 266	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( B +e -e A ) = ( B - A ) )
6	5	breq2d 3936	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( B +e -e A ) <-> 0 < ( B - A ) ) )
7	3, 6	bitr4d 190	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
8	7	ex 114	. . . . . 6 |- ( A e. RR -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
9		rexr 7804	. . . . . . . . . 10 |- ( B e. RR -> B e. RR* )
10		pnfnlt 9566	. . . . . . . . . . 11 |- ( B e. RR* -> -. +oo < B )
11	10	adantl 275	. . . . . . . . . 10 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
12	9, 11	sylan2 284	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> -. +oo < B )
13		simpl 108	. . . . . . . . . 10 |- ( ( A = +oo /\ B e. RR ) -> A = +oo )
14	13	breq1d 3934	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> ( A < B <-> +oo < B ) )
15	12, 14	mtbird 662	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR ) -> -. A < B )
16		0xr 7805	. . . . . . . . . 10 |- 0 e. RR*
17		nltmnf 9567	. . . . . . . . . 10 |- ( 0 e. RR* -> -. 0 < -oo )
18	16, 17	ax-mp 5	. . . . . . . . 9 |- -. 0 < -oo
19		xnegeq 9603	. . . . . . . . . . . . . 14 |- ( A = +oo -> -e A = -e +oo )
20	19	adantr 274	. . . . . . . . . . . . 13 |- ( ( A = +oo /\ B e. RR ) -> -e A = -e +oo )
21		xnegpnf 9604	. . . . . . . . . . . . 13 |- -e +oo = -oo
22	20, 21	syl6eq 2186	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B e. RR ) -> -e A = -oo )
23	22	oveq2d 5783	. . . . . . . . . . 11 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -e A ) = ( B +e -oo ) )
24		renepnf 7806	. . . . . . . . . . . . 13 |- ( B e. RR -> B =/= +oo )
25	24	adantl 275	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B e. RR ) -> B =/= +oo )
26		xaddmnf1 9624	. . . . . . . . . . . 12 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
27	9, 25, 26	syl2an2 583	. . . . . . . . . . 11 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -oo ) = -oo )
28	23, 27	eqtrd 2170	. . . . . . . . . 10 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -e A ) = -oo )
29	28	breq2d 3936	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
30	18, 29	mtbiri 664	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR ) -> -. 0 < ( B +e -e A ) )
31	15, 30	2falsed 691	. . . . . . 7 |- ( ( A = +oo /\ B e. RR ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
32	31	ex 114	. . . . . 6 |- ( A = +oo -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
33		simpl 108	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> A = -oo )
34		mnflt 9562	. . . . . . . . . 10 |- ( B e. RR -> -oo < B )
35	34	adantl 275	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> -oo < B )
36	33, 35	eqbrtrd 3945	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> A < B )
37		0ltpnf 9561	. . . . . . . . 9 |- 0 < +oo
38		xnegeq 9603	. . . . . . . . . . . . 13 |- ( A = -oo -> -e A = -e -oo )
39		xnegmnf 9605	. . . . . . . . . . . . 13 |- -e -oo = +oo
40	38, 39	syl6eq 2186	. . . . . . . . . . . 12 |- ( A = -oo -> -e A = +oo )
41	40	oveq2d 5783	. . . . . . . . . . 11 |- ( A = -oo -> ( B +e -e A ) = ( B +e +oo ) )
42	41	adantr 274	. . . . . . . . . 10 |- ( ( A = -oo /\ B e. RR ) -> ( B +e -e A ) = ( B +e +oo ) )
43		renemnf 7807	. . . . . . . . . . . 12 |- ( B e. RR -> B =/= -oo )
44	43	adantl 275	. . . . . . . . . . 11 |- ( ( A = -oo /\ B e. RR ) -> B =/= -oo )
45		xaddpnf1 9622	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
46	9, 44, 45	syl2an2 583	. . . . . . . . . 10 |- ( ( A = -oo /\ B e. RR ) -> ( B +e +oo ) = +oo )
47	42, 46	eqtrd 2170	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B +e -e A ) = +oo )
48	37, 47	breqtrrid 3961	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> 0 < ( B +e -e A ) )
49	36, 48	2thd 174	. . . . . . 7 |- ( ( A = -oo /\ B e. RR ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
50	49	ex 114	. . . . . 6 |- ( A = -oo -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
51	8, 32, 50	3jaoi 1281	. . . . 5 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
52	2, 51	sylbi 120	. . . 4 |- ( A e. RR* -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
53		ltpnf 9560	. . . . . . . . . 10 |- ( A e. RR -> A < +oo )
54	53	adantr 274	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A < +oo )
55		simpr 109	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> B = +oo )
56	54, 55	breqtrrd 3951	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> A < B )
57	55	oveq1d 5782	. . . . . . . . . 10 |- ( ( A e. RR /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e -e A ) )
58		rexneg 9606	. . . . . . . . . . . . . 14 |- ( A e. RR -> -e A = -u A )
59		renegcl 8016	. . . . . . . . . . . . . 14 |- ( A e. RR -> -u A e. RR )
60	58, 59	eqeltrd 2214	. . . . . . . . . . . . 13 |- ( A e. RR -> -e A e. RR )
61	60	rexrd 7808	. . . . . . . . . . . 12 |- ( A e. RR -> -e A e. RR* )
62	61	adantr 274	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = +oo ) -> -e A e. RR* )
63	60	renemnfd 7810	. . . . . . . . . . . 12 |- ( A e. RR -> -e A =/= -oo )
64	63	adantr 274	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = +oo ) -> -e A =/= -oo )
65		xaddpnf2 9623	. . . . . . . . . . 11 |- ( ( -e A e. RR* /\ -e A =/= -oo ) -> ( +oo +e -e A ) = +oo )
66	62, 64, 65	syl2anc 408	. . . . . . . . . 10 |- ( ( A e. RR /\ B = +oo ) -> ( +oo +e -e A ) = +oo )
67	57, 66	eqtrd 2170	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> ( B +e -e A ) = +oo )
68	37, 67	breqtrrid 3961	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> 0 < ( B +e -e A ) )
69	56, 68	2thd 174	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
70	69	ex 114	. . . . . 6 |- ( A e. RR -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
71		pnfxr 7811	. . . . . . . . . 10 |- +oo e. RR*
72		xrltnr 9559	. . . . . . . . . 10 |- ( +oo e. RR* -> -. +oo < +oo )
73	71, 72	ax-mp 5	. . . . . . . . 9 |- -. +oo < +oo
74		breq12 3929	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> +oo < +oo ) )
75	73, 74	mtbiri 664	. . . . . . . 8 |- ( ( A = +oo /\ B = +oo ) -> -. A < B )
76		0re 7759	. . . . . . . . . 10 |- 0 e. RR
77	76	ltnri 7849	. . . . . . . . 9 |- -. 0 < 0
78		simpr 109	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = +oo ) -> B = +oo )
79	19, 21	syl6eq 2186	. . . . . . . . . . . . 13 |- ( A = +oo -> -e A = -oo )
80	79	adantr 274	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = +oo ) -> -e A = -oo )
81	78, 80	oveq12d 5785	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e -oo ) )
82		pnfaddmnf 9626	. . . . . . . . . . 11 |- ( +oo +e -oo ) = 0
83	81, 82	syl6eq 2186	. . . . . . . . . 10 |- ( ( A = +oo /\ B = +oo ) -> ( B +e -e A ) = 0 )
84	83	breq2d 3936	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( 0 < ( B +e -e A ) <-> 0 < 0 ) )
85	77, 84	mtbiri 664	. . . . . . . 8 |- ( ( A = +oo /\ B = +oo ) -> -. 0 < ( B +e -e A ) )
86	75, 85	2falsed 691	. . . . . . 7 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
87	86	ex 114	. . . . . 6 |- ( A = +oo -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
88		mnfltpnf 9564	. . . . . . . . 9 |- -oo < +oo
89		breq12 3929	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
90	88, 89	mpbiri 167	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A < B )
91		oveq1 5774	. . . . . . . . . . 11 |- ( B = +oo -> ( B +e +oo ) = ( +oo +e +oo ) )
92	41, 91	sylan9eq 2190	. . . . . . . . . 10 |- ( ( A = -oo /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e +oo ) )
93		pnfnemnf 7813	. . . . . . . . . . 11 |- +oo =/= -oo
94		xaddpnf1 9622	. . . . . . . . . . 11 |- ( ( +oo e. RR* /\ +oo =/= -oo ) -> ( +oo +e +oo ) = +oo )
95	71, 93, 94	mp2an 422	. . . . . . . . . 10 |- ( +oo +e +oo ) = +oo
96	92, 95	syl6eq 2186	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( B +e -e A ) = +oo )
97	37, 96	breqtrrid 3961	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> 0 < ( B +e -e A ) )
98	90, 97	2thd 174	. . . . . . 7 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
99	98	ex 114	. . . . . 6 |- ( A = -oo -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
100	70, 87, 99	3jaoi 1281	. . . . 5 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
101	2, 100	sylbi 120	. . . 4 |- ( A e. RR* -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
102		rexr 7804	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
103	102	adantr 274	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> A e. RR* )
104		nltmnf 9567	. . . . . . . . . 10 |- ( A e. RR* -> -. A < -oo )
105	103, 104	syl 14	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
106		simpr 109	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> B = -oo )
107	106	breq2d 3936	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
108	105, 107	mtbird 662	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
109	106	oveq1d 5782	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = -oo ) -> ( B +e -e A ) = ( -oo +e -e A ) )
110		rexr 7804	. . . . . . . . . . . . . 14 |- ( -e A e. RR -> -e A e. RR* )
111		renepnf 7806	. . . . . . . . . . . . . 14 |- ( -e A e. RR -> -e A =/= +oo )
112		xaddmnf2 9625	. . . . . . . . . . . . . 14 |- ( ( -e A e. RR* /\ -e A =/= +oo ) -> ( -oo +e -e A ) = -oo )
113	110, 111, 112	syl2anc 408	. . . . . . . . . . . . 13 |- ( -e A e. RR -> ( -oo +e -e A ) = -oo )
114	60, 113	syl 14	. . . . . . . . . . . 12 |- ( A e. RR -> ( -oo +e -e A ) = -oo )
115	114	adantr 274	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = -oo ) -> ( -oo +e -e A ) = -oo )
116	109, 115	eqtrd 2170	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> ( B +e -e A ) = -oo )
117	116	breq2d 3936	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
118	18, 117	mtbiri 664	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
119	108, 118	2falsed 691	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
120	119	ex 114	. . . . . 6 |- ( A e. RR -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
121		eleq1 2200	. . . . . . . . . . . 12 |- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
122	71, 121	mpbiri 167	. . . . . . . . . . 11 |- ( A = +oo -> A e. RR* )
123	122	adantr 274	. . . . . . . . . 10 |- ( ( A = +oo /\ B = -oo ) -> A e. RR* )
124	123, 104	syl 14	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> -. A < -oo )
125		simpr 109	. . . . . . . . . 10 |- ( ( A = +oo /\ B = -oo ) -> B = -oo )
126	125	breq2d 3936	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> ( A < B <-> A < -oo ) )
127	124, 126	mtbird 662	. . . . . . . 8 |- ( ( A = +oo /\ B = -oo ) -> -. A < B )
128	79	oveq2d 5783	. . . . . . . . . . . 12 |- ( A = +oo -> ( B +e -e A ) = ( B +e -oo ) )
129	128	adantr 274	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -e A ) = ( B +e -oo ) )
130		mnfxr 7815	. . . . . . . . . . . . 13 |- -oo e. RR*
131		eleq1 2200	. . . . . . . . . . . . 13 |- ( B = -oo -> ( B e. RR* <-> -oo e. RR* ) )
132	130, 131	mpbiri 167	. . . . . . . . . . . 12 |- ( B = -oo -> B e. RR* )
133		mnfnepnf 7814	. . . . . . . . . . . . . 14 |- -oo =/= +oo
134		neeq1 2319	. . . . . . . . . . . . . 14 |- ( B = -oo -> ( B =/= +oo <-> -oo =/= +oo ) )
135	133, 134	mpbiri 167	. . . . . . . . . . . . 13 |- ( B = -oo -> B =/= +oo )
136	135	adantl 275	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = -oo ) -> B =/= +oo )
137	132, 136, 26	syl2an2 583	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -oo ) = -oo )
138	129, 137	eqtrd 2170	. . . . . . . . . 10 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -e A ) = -oo )
139	138	breq2d 3936	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
140	18, 139	mtbiri 664	. . . . . . . 8 |- ( ( A = +oo /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
141	127, 140	2falsed 691	. . . . . . 7 |- ( ( A = +oo /\ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
142	141	ex 114	. . . . . 6 |- ( A = +oo -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
143		xrltnr 9559	. . . . . . . . . 10 |- ( -oo e. RR* -> -. -oo < -oo )
144	130, 143	ax-mp 5	. . . . . . . . 9 |- -. -oo < -oo
145		breq12 3929	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
146	144, 145	mtbiri 664	. . . . . . . 8 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
147		oveq1 5774	. . . . . . . . . . . 12 |- ( B = -oo -> ( B +e +oo ) = ( -oo +e +oo ) )
148	41, 147	sylan9eq 2190	. . . . . . . . . . 11 |- ( ( A = -oo /\ B = -oo ) -> ( B +e -e A ) = ( -oo +e +oo ) )
149		mnfaddpnf 9627	. . . . . . . . . . 11 |- ( -oo +e +oo ) = 0
150	148, 149	syl6eq 2186	. . . . . . . . . 10 |- ( ( A = -oo /\ B = -oo ) -> ( B +e -e A ) = 0 )
151	150	breq2d 3936	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < 0 ) )
152	77, 151	mtbiri 664	. . . . . . . 8 |- ( ( A = -oo /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
153	146, 152	2falsed 691	. . . . . . 7 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
154	153	ex 114	. . . . . 6 |- ( A = -oo -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
155	120, 142, 154	3jaoi 1281	. . . . 5 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
156	2, 155	sylbi 120	. . . 4 |- ( A e. RR* -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
157	52, 101, 156	3jaod 1282	. . 3 |- ( A e. RR* -> ( ( B e. RR \/ B = +oo \/ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
158	1, 157	syl5bi 151	. 2 |- ( A e. RR* -> ( B e. RR* -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
159	158	imp 123	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> 0 < ( B +e -e A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 961   = wceq 1331   e. wcel 1480   =/= wne 2306   class class class wbr 3924  (class class class)co 5767   RRcr 7612   0cc0 7613   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   < clt 7793   - cmin 7926   -ucneg 7927   -ecxne 9549   +ecxad 9550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-sub 7928  df-neg 7929  df-xneg 9552  df-xadd 9553

This theorem is referenced by: (None)