Theorem xposdif 10284

Description: Extended real version of posdif 8783. (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 10178	. . 3 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		elxr 10178	. . . . 5 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
3		posdif 8783	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
4		rexsub 10255	. . . . . . . . . 10 |- ( ( B e. RR /\ A e. RR ) -> ( B +e -e A ) = ( B - A ) )
5	4	ancoms 268	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( B +e -e A ) = ( B - A ) )
6	5	breq2d 4142	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( B +e -e A ) <-> 0 < ( B - A ) ) )
7	3, 6	bitr4d 191	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
8	7	ex 115	. . . . . 6 |- ( A e. RR -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
9		rexr 8371	. . . . . . . . . 10 |- ( B e. RR -> B e. RR* )
10		pnfnlt 10189	. . . . . . . . . . 11 |- ( B e. RR* -> -. +oo < B )
11	10	adantl 277	. . . . . . . . . 10 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
12	9, 11	sylan2 286	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> -. +oo < B )
13		simpl 109	. . . . . . . . . 10 |- ( ( A = +oo /\ B e. RR ) -> A = +oo )
14	13	breq1d 4140	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> ( A < B <-> +oo < B ) )
15	12, 14	mtbird 684	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR ) -> -. A < B )
16		0xr 8372	. . . . . . . . . 10 |- 0 e. RR*
17		nltmnf 10190	. . . . . . . . . 10 |- ( 0 e. RR* -> -. 0 < -oo )
18	16, 17	ax-mp 5	. . . . . . . . 9 |- -. 0 < -oo
19		xnegeq 10229	. . . . . . . . . . . . . 14 |- ( A = +oo -> -e A = -e +oo )
20	19	adantr 276	. . . . . . . . . . . . 13 |- ( ( A = +oo /\ B e. RR ) -> -e A = -e +oo )
21		xnegpnf 10230	. . . . . . . . . . . . 13 |- -e +oo = -oo
22	20, 21	eqtrdi 2287	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B e. RR ) -> -e A = -oo )
23	22	oveq2d 6101	. . . . . . . . . . 11 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -e A ) = ( B +e -oo ) )
24		renepnf 8373	. . . . . . . . . . . . 13 |- ( B e. RR -> B =/= +oo )
25	24	adantl 277	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B e. RR ) -> B =/= +oo )
26		xaddmnf1 10250	. . . . . . . . . . . 12 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
27	9, 25, 26	syl2an2 602	. . . . . . . . . . 11 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -oo ) = -oo )
28	23, 27	eqtrd 2271	. . . . . . . . . 10 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -e A ) = -oo )
29	28	breq2d 4142	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
30	18, 29	mtbiri 686	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR ) -> -. 0 < ( B +e -e A ) )
31	15, 30	2falsed 714	. . . . . . 7 |- ( ( A = +oo /\ B e. RR ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
32	31	ex 115	. . . . . 6 |- ( A = +oo -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
33		simpl 109	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> A = -oo )
34		mnflt 10185	. . . . . . . . . 10 |- ( B e. RR -> -oo < B )
35	34	adantl 277	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> -oo < B )
36	33, 35	eqbrtrd 4152	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> A < B )
37		0ltpnf 10184	. . . . . . . . 9 |- 0 < +oo
38		xnegeq 10229	. . . . . . . . . . . . 13 |- ( A = -oo -> -e A = -e -oo )
39		xnegmnf 10231	. . . . . . . . . . . . 13 |- -e -oo = +oo
40	38, 39	eqtrdi 2287	. . . . . . . . . . . 12 |- ( A = -oo -> -e A = +oo )
41	40	oveq2d 6101	. . . . . . . . . . 11 |- ( A = -oo -> ( B +e -e A ) = ( B +e +oo ) )
42	41	adantr 276	. . . . . . . . . 10 |- ( ( A = -oo /\ B e. RR ) -> ( B +e -e A ) = ( B +e +oo ) )
43		renemnf 8374	. . . . . . . . . . . 12 |- ( B e. RR -> B =/= -oo )
44	43	adantl 277	. . . . . . . . . . 11 |- ( ( A = -oo /\ B e. RR ) -> B =/= -oo )
45		xaddpnf1 10248	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
46	9, 44, 45	syl2an2 602	. . . . . . . . . 10 |- ( ( A = -oo /\ B e. RR ) -> ( B +e +oo ) = +oo )
47	42, 46	eqtrd 2271	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B +e -e A ) = +oo )
48	37, 47	breqtrrid 4168	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> 0 < ( B +e -e A ) )
49	36, 48	2thd 175	. . . . . . 7 |- ( ( A = -oo /\ B e. RR ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
50	49	ex 115	. . . . . 6 |- ( A = -oo -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
51	8, 32, 50	3jaoi 1344	. . . . 5 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
52	2, 51	sylbi 121	. . . 4 |- ( A e. RR* -> ( B e. RR -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
53		ltpnf 10182	. . . . . . . . . 10 |- ( A e. RR -> A < +oo )
54	53	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A < +oo )
55		simpr 110	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> B = +oo )
56	54, 55	breqtrrd 4158	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> A < B )
57	55	oveq1d 6100	. . . . . . . . . 10 |- ( ( A e. RR /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e -e A ) )
58		rexneg 10232	. . . . . . . . . . . . . 14 |- ( A e. RR -> -e A = -u A )
59		renegcl 8587	. . . . . . . . . . . . . 14 |- ( A e. RR -> -u A e. RR )
60	58, 59	eqeltrd 2315	. . . . . . . . . . . . 13 |- ( A e. RR -> -e A e. RR )
61	60	rexrd 8375	. . . . . . . . . . . 12 |- ( A e. RR -> -e A e. RR* )
62	61	adantr 276	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = +oo ) -> -e A e. RR* )
63	60	renemnfd 8377	. . . . . . . . . . . 12 |- ( A e. RR -> -e A =/= -oo )
64	63	adantr 276	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = +oo ) -> -e A =/= -oo )
65		xaddpnf2 10249	. . . . . . . . . . 11 |- ( ( -e A e. RR* /\ -e A =/= -oo ) -> ( +oo +e -e A ) = +oo )
66	62, 64, 65	syl2anc 415	. . . . . . . . . 10 |- ( ( A e. RR /\ B = +oo ) -> ( +oo +e -e A ) = +oo )
67	57, 66	eqtrd 2271	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> ( B +e -e A ) = +oo )
68	37, 67	breqtrrid 4168	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> 0 < ( B +e -e A ) )
69	56, 68	2thd 175	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
70	69	ex 115	. . . . . 6 |- ( A e. RR -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
71		pnfxr 8378	. . . . . . . . . 10 |- +oo e. RR*
72		xrltnr 10181	. . . . . . . . . 10 |- ( +oo e. RR* -> -. +oo < +oo )
73	71, 72	ax-mp 5	. . . . . . . . 9 |- -. +oo < +oo
74		breq12 4135	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> +oo < +oo ) )
75	73, 74	mtbiri 686	. . . . . . . 8 |- ( ( A = +oo /\ B = +oo ) -> -. A < B )
76		0re 8326	. . . . . . . . . 10 |- 0 e. RR
77	76	ltnri 8418	. . . . . . . . 9 |- -. 0 < 0
78		simpr 110	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = +oo ) -> B = +oo )
79	19, 21	eqtrdi 2287	. . . . . . . . . . . . 13 |- ( A = +oo -> -e A = -oo )
80	79	adantr 276	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = +oo ) -> -e A = -oo )
81	78, 80	oveq12d 6103	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e -oo ) )
82		pnfaddmnf 10252	. . . . . . . . . . 11 |- ( +oo +e -oo ) = 0
83	81, 82	eqtrdi 2287	. . . . . . . . . 10 |- ( ( A = +oo /\ B = +oo ) -> ( B +e -e A ) = 0 )
84	83	breq2d 4142	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( 0 < ( B +e -e A ) <-> 0 < 0 ) )
85	77, 84	mtbiri 686	. . . . . . . 8 |- ( ( A = +oo /\ B = +oo ) -> -. 0 < ( B +e -e A ) )
86	75, 85	2falsed 714	. . . . . . 7 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
87	86	ex 115	. . . . . 6 |- ( A = +oo -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
88		mnfltpnf 10187	. . . . . . . . 9 |- -oo < +oo
89		breq12 4135	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
90	88, 89	mpbiri 168	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A < B )
91		oveq1 6092	. . . . . . . . . . 11 |- ( B = +oo -> ( B +e +oo ) = ( +oo +e +oo ) )
92	41, 91	sylan9eq 2291	. . . . . . . . . 10 |- ( ( A = -oo /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e +oo ) )
93		pnfnemnf 8380	. . . . . . . . . . 11 |- +oo =/= -oo
94		xaddpnf1 10248	. . . . . . . . . . 11 |- ( ( +oo e. RR* /\ +oo =/= -oo ) -> ( +oo +e +oo ) = +oo )
95	71, 93, 94	mp2an 430	. . . . . . . . . 10 |- ( +oo +e +oo ) = +oo
96	92, 95	eqtrdi 2287	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( B +e -e A ) = +oo )
97	37, 96	breqtrrid 4168	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> 0 < ( B +e -e A ) )
98	90, 97	2thd 175	. . . . . . 7 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
99	98	ex 115	. . . . . 6 |- ( A = -oo -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
100	70, 87, 99	3jaoi 1344	. . . . 5 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
101	2, 100	sylbi 121	. . . 4 |- ( A e. RR* -> ( B = +oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
102		rexr 8371	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
103	102	adantr 276	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> A e. RR* )
104		nltmnf 10190	. . . . . . . . . 10 |- ( A e. RR* -> -. A < -oo )
105	103, 104	syl 14	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
106		simpr 110	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> B = -oo )
107	106	breq2d 4142	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
108	105, 107	mtbird 684	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
109	106	oveq1d 6100	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = -oo ) -> ( B +e -e A ) = ( -oo +e -e A ) )
110		rexr 8371	. . . . . . . . . . . . . 14 |- ( -e A e. RR -> -e A e. RR* )
111		renepnf 8373	. . . . . . . . . . . . . 14 |- ( -e A e. RR -> -e A =/= +oo )
112		xaddmnf2 10251	. . . . . . . . . . . . . 14 |- ( ( -e A e. RR* /\ -e A =/= +oo ) -> ( -oo +e -e A ) = -oo )
113	110, 111, 112	syl2anc 415	. . . . . . . . . . . . 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 276	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = -oo ) -> ( -oo +e -e A ) = -oo )
116	109, 115	eqtrd 2271	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> ( B +e -e A ) = -oo )
117	116	breq2d 4142	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
118	18, 117	mtbiri 686	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
119	108, 118	2falsed 714	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
120	119	ex 115	. . . . . 6 |- ( A e. RR -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
121		eleq1 2301	. . . . . . . . . . . 12 |- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
122	71, 121	mpbiri 168	. . . . . . . . . . 11 |- ( A = +oo -> A e. RR* )
123	122	adantr 276	. . . . . . . . . 10 |- ( ( A = +oo /\ B = -oo ) -> A e. RR* )
124	123, 104	syl 14	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> -. A < -oo )
125		simpr 110	. . . . . . . . . 10 |- ( ( A = +oo /\ B = -oo ) -> B = -oo )
126	125	breq2d 4142	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> ( A < B <-> A < -oo ) )
127	124, 126	mtbird 684	. . . . . . . 8 |- ( ( A = +oo /\ B = -oo ) -> -. A < B )
128	79	oveq2d 6101	. . . . . . . . . . . 12 |- ( A = +oo -> ( B +e -e A ) = ( B +e -oo ) )
129	128	adantr 276	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -e A ) = ( B +e -oo ) )
130		mnfxr 8382	. . . . . . . . . . . . 13 |- -oo e. RR*
131		eleq1 2301	. . . . . . . . . . . . 13 |- ( B = -oo -> ( B e. RR* <-> -oo e. RR* ) )
132	130, 131	mpbiri 168	. . . . . . . . . . . 12 |- ( B = -oo -> B e. RR* )
133		mnfnepnf 8381	. . . . . . . . . . . . . 14 |- -oo =/= +oo
134		neeq1 2433	. . . . . . . . . . . . . 14 |- ( B = -oo -> ( B =/= +oo <-> -oo =/= +oo ) )
135	133, 134	mpbiri 168	. . . . . . . . . . . . 13 |- ( B = -oo -> B =/= +oo )
136	135	adantl 277	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = -oo ) -> B =/= +oo )
137	132, 136, 26	syl2an2 602	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -oo ) = -oo )
138	129, 137	eqtrd 2271	. . . . . . . . . 10 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -e A ) = -oo )
139	138	breq2d 4142	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
140	18, 139	mtbiri 686	. . . . . . . 8 |- ( ( A = +oo /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
141	127, 140	2falsed 714	. . . . . . 7 |- ( ( A = +oo /\ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
142	141	ex 115	. . . . . 6 |- ( A = +oo -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
143		xrltnr 10181	. . . . . . . . . 10 |- ( -oo e. RR* -> -. -oo < -oo )
144	130, 143	ax-mp 5	. . . . . . . . 9 |- -. -oo < -oo
145		breq12 4135	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
146	144, 145	mtbiri 686	. . . . . . . 8 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
147		oveq1 6092	. . . . . . . . . . . 12 |- ( B = -oo -> ( B +e +oo ) = ( -oo +e +oo ) )
148	41, 147	sylan9eq 2291	. . . . . . . . . . 11 |- ( ( A = -oo /\ B = -oo ) -> ( B +e -e A ) = ( -oo +e +oo ) )
149		mnfaddpnf 10253	. . . . . . . . . . 11 |- ( -oo +e +oo ) = 0
150	148, 149	eqtrdi 2287	. . . . . . . . . 10 |- ( ( A = -oo /\ B = -oo ) -> ( B +e -e A ) = 0 )
151	150	breq2d 4142	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < 0 ) )
152	77, 151	mtbiri 686	. . . . . . . 8 |- ( ( A = -oo /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
153	146, 152	2falsed 714	. . . . . . 7 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
154	153	ex 115	. . . . . 6 |- ( A = -oo -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
155	120, 142, 154	3jaoi 1344	. . . . 5 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
156	2, 155	sylbi 121	. . . 4 |- ( A e. RR* -> ( B = -oo -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
157	52, 101, 156	3jaod 1345	. . 3 |- ( A e. RR* -> ( ( B e. RR \/ B = +oo \/ B = -oo ) -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
158	1, 157	biimtrid 152	. 2 |- ( A e. RR* -> ( B e. RR* -> ( A < B <-> 0 < ( B +e -e A ) ) ) )
159	158	imp 124	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> 0 < ( B +e -e A ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1008   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   < clt 8360   - cmin 8497   -ucneg 8498   -ecxne 10171   +ecxad 10172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-sub 8499  df-neg 8500  df-xneg 10174  df-xadd 10175

This theorem is used by: (None)