Theorem xposdif 9818

Description: Extended real version of posdif 8353. (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 9712	. . 3 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
2		elxr 9712	. . . . 5 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
3		posdif 8353	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
4		rexsub 9789	. . . . . . . . . 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 3994	. . . . . . . 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 7944	. . . . . . . . . 10 |- ( B e. RR -> B e. RR* )
10		pnfnlt 9723	. . . . . . . . . . 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 3992	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> ( A < B <-> +oo < B ) )
15	12, 14	mtbird 663	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR ) -> -. A < B )
16		0xr 7945	. . . . . . . . . 10 |- 0 e. RR*
17		nltmnf 9724	. . . . . . . . . 10 |- ( 0 e. RR* -> -. 0 < -oo )
18	16, 17	ax-mp 5	. . . . . . . . 9 |- -. 0 < -oo
19		xnegeq 9763	. . . . . . . . . . . . . 14 |- ( A = +oo -> -e A = -e +oo )
20	19	adantr 274	. . . . . . . . . . . . 13 |- ( ( A = +oo /\ B e. RR ) -> -e A = -e +oo )
21		xnegpnf 9764	. . . . . . . . . . . . 13 |- -e +oo = -oo
22	20, 21	eqtrdi 2215	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B e. RR ) -> -e A = -oo )
23	22	oveq2d 5858	. . . . . . . . . . 11 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -e A ) = ( B +e -oo ) )
24		renepnf 7946	. . . . . . . . . . . . 13 |- ( B e. RR -> B =/= +oo )
25	24	adantl 275	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B e. RR ) -> B =/= +oo )
26		xaddmnf1 9784	. . . . . . . . . . . 12 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
27	9, 25, 26	syl2an2 584	. . . . . . . . . . 11 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -oo ) = -oo )
28	23, 27	eqtrd 2198	. . . . . . . . . 10 |- ( ( A = +oo /\ B e. RR ) -> ( B +e -e A ) = -oo )
29	28	breq2d 3994	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
30	18, 29	mtbiri 665	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR ) -> -. 0 < ( B +e -e A ) )
31	15, 30	2falsed 692	. . . . . . 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 9719	. . . . . . . . . 10 |- ( B e. RR -> -oo < B )
35	34	adantl 275	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> -oo < B )
36	33, 35	eqbrtrd 4004	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> A < B )
37		0ltpnf 9718	. . . . . . . . 9 |- 0 < +oo
38		xnegeq 9763	. . . . . . . . . . . . 13 |- ( A = -oo -> -e A = -e -oo )
39		xnegmnf 9765	. . . . . . . . . . . . 13 |- -e -oo = +oo
40	38, 39	eqtrdi 2215	. . . . . . . . . . . 12 |- ( A = -oo -> -e A = +oo )
41	40	oveq2d 5858	. . . . . . . . . . 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 7947	. . . . . . . . . . . 12 |- ( B e. RR -> B =/= -oo )
44	43	adantl 275	. . . . . . . . . . 11 |- ( ( A = -oo /\ B e. RR ) -> B =/= -oo )
45		xaddpnf1 9782	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
46	9, 44, 45	syl2an2 584	. . . . . . . . . 10 |- ( ( A = -oo /\ B e. RR ) -> ( B +e +oo ) = +oo )
47	42, 46	eqtrd 2198	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B +e -e A ) = +oo )
48	37, 47	breqtrrid 4020	. . . . . . . 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 1293	. . . . 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 9716	. . . . . . . . . 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 4010	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> A < B )
57	55	oveq1d 5857	. . . . . . . . . 10 |- ( ( A e. RR /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e -e A ) )
58		rexneg 9766	. . . . . . . . . . . . . 14 |- ( A e. RR -> -e A = -u A )
59		renegcl 8159	. . . . . . . . . . . . . 14 |- ( A e. RR -> -u A e. RR )
60	58, 59	eqeltrd 2243	. . . . . . . . . . . . 13 |- ( A e. RR -> -e A e. RR )
61	60	rexrd 7948	. . . . . . . . . . . 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 7950	. . . . . . . . . . . 12 |- ( A e. RR -> -e A =/= -oo )
64	63	adantr 274	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = +oo ) -> -e A =/= -oo )
65		xaddpnf2 9783	. . . . . . . . . . 11 |- ( ( -e A e. RR* /\ -e A =/= -oo ) -> ( +oo +e -e A ) = +oo )
66	62, 64, 65	syl2anc 409	. . . . . . . . . 10 |- ( ( A e. RR /\ B = +oo ) -> ( +oo +e -e A ) = +oo )
67	57, 66	eqtrd 2198	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> ( B +e -e A ) = +oo )
68	37, 67	breqtrrid 4020	. . . . . . . 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 7951	. . . . . . . . . 10 |- +oo e. RR*
72		xrltnr 9715	. . . . . . . . . 10 |- ( +oo e. RR* -> -. +oo < +oo )
73	71, 72	ax-mp 5	. . . . . . . . 9 |- -. +oo < +oo
74		breq12 3987	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> +oo < +oo ) )
75	73, 74	mtbiri 665	. . . . . . . 8 |- ( ( A = +oo /\ B = +oo ) -> -. A < B )
76		0re 7899	. . . . . . . . . 10 |- 0 e. RR
77	76	ltnri 7991	. . . . . . . . 9 |- -. 0 < 0
78		simpr 109	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = +oo ) -> B = +oo )
79	19, 21	eqtrdi 2215	. . . . . . . . . . . . 13 |- ( A = +oo -> -e A = -oo )
80	79	adantr 274	. . . . . . . . . . . 12 |- ( ( A = +oo /\ B = +oo ) -> -e A = -oo )
81	78, 80	oveq12d 5860	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e -oo ) )
82		pnfaddmnf 9786	. . . . . . . . . . 11 |- ( +oo +e -oo ) = 0
83	81, 82	eqtrdi 2215	. . . . . . . . . 10 |- ( ( A = +oo /\ B = +oo ) -> ( B +e -e A ) = 0 )
84	83	breq2d 3994	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( 0 < ( B +e -e A ) <-> 0 < 0 ) )
85	77, 84	mtbiri 665	. . . . . . . 8 |- ( ( A = +oo /\ B = +oo ) -> -. 0 < ( B +e -e A ) )
86	75, 85	2falsed 692	. . . . . . 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 9721	. . . . . . . . 9 |- -oo < +oo
89		breq12 3987	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
90	88, 89	mpbiri 167	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A < B )
91		oveq1 5849	. . . . . . . . . . 11 |- ( B = +oo -> ( B +e +oo ) = ( +oo +e +oo ) )
92	41, 91	sylan9eq 2219	. . . . . . . . . 10 |- ( ( A = -oo /\ B = +oo ) -> ( B +e -e A ) = ( +oo +e +oo ) )
93		pnfnemnf 7953	. . . . . . . . . . 11 |- +oo =/= -oo
94		xaddpnf1 9782	. . . . . . . . . . 11 |- ( ( +oo e. RR* /\ +oo =/= -oo ) -> ( +oo +e +oo ) = +oo )
95	71, 93, 94	mp2an 423	. . . . . . . . . 10 |- ( +oo +e +oo ) = +oo
96	92, 95	eqtrdi 2215	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( B +e -e A ) = +oo )
97	37, 96	breqtrrid 4020	. . . . . . . 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 1293	. . . . 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 7944	. . . . . . . . . . 11 |- ( A e. RR -> A e. RR* )
103	102	adantr 274	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> A e. RR* )
104		nltmnf 9724	. . . . . . . . . 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 3994	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
108	105, 107	mtbird 663	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
109	106	oveq1d 5857	. . . . . . . . . . 11 |- ( ( A e. RR /\ B = -oo ) -> ( B +e -e A ) = ( -oo +e -e A ) )
110		rexr 7944	. . . . . . . . . . . . . 14 |- ( -e A e. RR -> -e A e. RR* )
111		renepnf 7946	. . . . . . . . . . . . . 14 |- ( -e A e. RR -> -e A =/= +oo )
112		xaddmnf2 9785	. . . . . . . . . . . . . 14 |- ( ( -e A e. RR* /\ -e A =/= +oo ) -> ( -oo +e -e A ) = -oo )
113	110, 111, 112	syl2anc 409	. . . . . . . . . . . . 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 2198	. . . . . . . . . 10 |- ( ( A e. RR /\ B = -oo ) -> ( B +e -e A ) = -oo )
117	116	breq2d 3994	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
118	18, 117	mtbiri 665	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
119	108, 118	2falsed 692	. . . . . . 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 2229	. . . . . . . . . . . 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 3994	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> ( A < B <-> A < -oo ) )
127	124, 126	mtbird 663	. . . . . . . 8 |- ( ( A = +oo /\ B = -oo ) -> -. A < B )
128	79	oveq2d 5858	. . . . . . . . . . . 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 7955	. . . . . . . . . . . . 13 |- -oo e. RR*
131		eleq1 2229	. . . . . . . . . . . . 13 |- ( B = -oo -> ( B e. RR* <-> -oo e. RR* ) )
132	130, 131	mpbiri 167	. . . . . . . . . . . 12 |- ( B = -oo -> B e. RR* )
133		mnfnepnf 7954	. . . . . . . . . . . . . 14 |- -oo =/= +oo
134		neeq1 2349	. . . . . . . . . . . . . 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 584	. . . . . . . . . . 11 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -oo ) = -oo )
138	129, 137	eqtrd 2198	. . . . . . . . . 10 |- ( ( A = +oo /\ B = -oo ) -> ( B +e -e A ) = -oo )
139	138	breq2d 3994	. . . . . . . . 9 |- ( ( A = +oo /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < -oo ) )
140	18, 139	mtbiri 665	. . . . . . . 8 |- ( ( A = +oo /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
141	127, 140	2falsed 692	. . . . . . 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 9715	. . . . . . . . . 10 |- ( -oo e. RR* -> -. -oo < -oo )
144	130, 143	ax-mp 5	. . . . . . . . 9 |- -. -oo < -oo
145		breq12 3987	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
146	144, 145	mtbiri 665	. . . . . . . 8 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
147		oveq1 5849	. . . . . . . . . . . 12 |- ( B = -oo -> ( B +e +oo ) = ( -oo +e +oo ) )
148	41, 147	sylan9eq 2219	. . . . . . . . . . 11 |- ( ( A = -oo /\ B = -oo ) -> ( B +e -e A ) = ( -oo +e +oo ) )
149		mnfaddpnf 9787	. . . . . . . . . . 11 |- ( -oo +e +oo ) = 0
150	148, 149	eqtrdi 2215	. . . . . . . . . 10 |- ( ( A = -oo /\ B = -oo ) -> ( B +e -e A ) = 0 )
151	150	breq2d 3994	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( 0 < ( B +e -e A ) <-> 0 < 0 ) )
152	77, 151	mtbiri 665	. . . . . . . 8 |- ( ( A = -oo /\ B = -oo ) -> -. 0 < ( B +e -e A ) )
153	146, 152	2falsed 692	. . . . . . 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 1293	. . . . 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 1294	. . 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 967   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933   - cmin 8069   -ucneg 8070   -ecxne 9705   +ecxad 9706

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-sub 8071  df-neg 8072  df-xneg 9708  df-xadd 9709

This theorem is referenced by: (None)