Theorem xrmaxltsup 11250

Description: Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)

Assertion

Ref	Expression
xrmaxltsup	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) < C <-> ( A < C /\ B < C ) ) )

Proof of Theorem xrmaxltsup

Step	Hyp	Ref	Expression
1		simpl1 1000	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A e. RR* )
2		simpl2 1001	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B e. RR* )
3		xrmaxcl 11244	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> sup ( { A , B } , RR* , < ) e. RR* )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> sup ( { A , B } , RR* , < ) e. RR* )
5		simpl3 1002	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> C e. RR* )
6		xrmax1sup 11245	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
7	6	3adant3 1017	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
8	7	adantr 276	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A <_ sup ( { A , B } , RR* , < ) )
9		simpr 110	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> sup ( { A , B } , RR* , < ) < C )
10	1, 4, 5, 8, 9	xrlelttrd 9797	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A < C )
11		xrmax2sup 11246	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ sup ( { A , B } , RR* , < ) )
12	1, 2, 11	syl2anc 411	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B <_ sup ( { A , B } , RR* , < ) )
13	2, 4, 5, 12, 9	xrlelttrd 9797	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B < C )
14	10, 13	jca 306	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> ( A < C /\ B < C ) )
15		simplr 528	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> A e. RR )
16		simpllr 534	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> B e. RR )
17		xrmaxrecl 11247	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR* , < ) = sup ( { A , B } , RR , < ) )
18	15, 16, 17	syl2anc 411	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> sup ( { A , B } , RR* , < ) = sup ( { A , B } , RR , < ) )
19		simp-4r 542	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> ( A < C /\ B < C ) )
20		simpr 110	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> C e. RR )
21		maxltsup 11211	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) < C <-> ( A < C /\ B < C ) ) )
22	15, 16, 20, 21	syl3anc 1238	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) < C <-> ( A < C /\ B < C ) ) )
23	19, 22	mpbird 167	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> sup ( { A , B } , RR , < ) < C )
24	18, 23	eqbrtrd 4022	. . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C e. RR ) -> sup ( { A , B } , RR* , < ) < C )
25		simplr 528	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> A e. RR )
26		simpllr 534	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> B e. RR )
27		maxcl 11203	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
28	25, 26, 27	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> sup ( { A , B } , RR , < ) e. RR )
29	17	eleq1d 2246	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { A , B } , RR* , < ) e. RR <-> sup ( { A , B } , RR , < ) e. RR ) )
30	25, 26, 29	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> ( sup ( { A , B } , RR* , < ) e. RR <-> sup ( { A , B } , RR , < ) e. RR ) )
31	28, 30	mpbird 167	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> sup ( { A , B } , RR* , < ) e. RR )
32		ltpnf 9767	. . . . . . 7 |- ( sup ( { A , B } , RR* , < ) e. RR -> sup ( { A , B } , RR* , < ) < +oo )
33	31, 32	syl 14	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> sup ( { A , B } , RR* , < ) < +oo )
34		simpr 110	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> C = +oo )
35	33, 34	breqtrrd 4028	. . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = +oo ) -> sup ( { A , B } , RR* , < ) < C )
36		simprl 529	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> A < C )
37	36	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = -oo ) -> A < C )
38		nltmnf 9775	. . . . . . . . 9 |- ( A e. RR* -> -. A < -oo )
39	38	3ad2ant1 1018	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. A < -oo )
40	39	ad4antr 494	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = -oo ) -> -. A < -oo )
41		simpr 110	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = -oo ) -> C = -oo )
42	41	breq2d 4012	. . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = -oo ) -> ( A < C <-> A < -oo ) )
43	40, 42	mtbird 673	. . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = -oo ) -> -. A < C )
44	37, 43	pm2.21dd 620	. . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) /\ C = -oo ) -> sup ( { A , B } , RR* , < ) < C )
45		elxr 9763	. . . . . . . 8 |- ( C e. RR* <-> ( C e. RR \/ C = +oo \/ C = -oo ) )
46	45	biimpi 120	. . . . . . 7 |- ( C e. RR* -> ( C e. RR \/ C = +oo \/ C = -oo ) )
47	46	3ad2ant3 1020	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. RR \/ C = +oo \/ C = -oo ) )
48	47	ad3antrrr 492	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) -> ( C e. RR \/ C = +oo \/ C = -oo ) )
49	24, 35, 44, 48	mpjao3dan 1307	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A e. RR ) -> sup ( { A , B } , RR* , < ) < C )
50	36	ad2antrr 488	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> A < C )
51		pnfnlt 9774	. . . . . . . 8 |- ( C e. RR* -> -. +oo < C )
52	51	3ad2ant3 1020	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. +oo < C )
53	52	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> -. +oo < C )
54		simpr 110	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> A = +oo )
55	54	breq1d 4010	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> ( A < C <-> +oo < C ) )
56	53, 55	mtbird 673	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> -. A < C )
57	50, 56	pm2.21dd 620	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> sup ( { A , B } , RR* , < ) < C )
58		simpr 110	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A = -oo )
59		mnfle 9779	. . . . . . . . 9 |- ( B e. RR* -> -oo <_ B )
60	59	3ad2ant2 1019	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -oo <_ B )
61	60	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> -oo <_ B )
62	58, 61	eqbrtrd 4022	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A <_ B )
63		simp1 997	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
64	63	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A e. RR* )
65		simp2 998	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
66	65	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> B e. RR* )
67		xrmaxleim 11236	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )
68	64, 66, 67	syl2anc 411	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )
69	62, 68	mpd 13	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> sup ( { A , B } , RR* , < ) = B )
70		simprr 531	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> B < C )
71	70	ad2antrr 488	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> B < C )
72	69, 71	eqbrtrd 4022	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> sup ( { A , B } , RR* , < ) < C )
73		elxr 9763	. . . . . . 7 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
74	73	biimpi 120	. . . . . 6 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
75	74	3ad2ant1 1018	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
76	75	ad2antrr 488	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
77	49, 57, 72, 76	mpjao3dan 1307	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) -> sup ( { A , B } , RR* , < ) < C )
78		simplrr 536	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> B < C )
79		breq1 4003	. . . . . 6 |- ( B = +oo -> ( B < C <-> +oo < C ) )
80	79	adantl 277	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> ( B < C <-> +oo < C ) )
81	78, 80	mpbid 147	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> +oo < C )
82	52	ad2antrr 488	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> -. +oo < C )
83	81, 82	pm2.21dd 620	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> sup ( { A , B } , RR* , < ) < C )
84		prcom 3667	. . . . . 6 |- { B , A } = { A , B }
85	84	supeq1i 6981	. . . . 5 |- sup ( { B , A } , RR* , < ) = sup ( { A , B } , RR* , < )
86		simpr 110	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B = -oo )
87		mnfle 9779	. . . . . . . . 9 |- ( A e. RR* -> -oo <_ A )
88	87	3ad2ant1 1018	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -oo <_ A )
89	88	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> -oo <_ A )
90	86, 89	eqbrtrd 4022	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B <_ A )
91		simpll2 1037	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B e. RR* )
92		simpll1 1036	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A e. RR* )
93		xrmaxleim 11236	. . . . . . 7 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A -> sup ( { B , A } , RR* , < ) = A ) )
94	91, 92, 93	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> ( B <_ A -> sup ( { B , A } , RR* , < ) = A ) )
95	90, 94	mpd 13	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { B , A } , RR* , < ) = A )
96	85, 95	eqtr3id 2224	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) = A )
97		simplrl 535	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A < C )
98	96, 97	eqbrtrd 4022	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) < C )
99		elxr 9763	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
100	99	biimpi 120	. . . . 5 |- ( B e. RR* -> ( B e. RR \/ B = +oo \/ B = -oo ) )
101	100	3ad2ant2 1019	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
102	101	adantr 276	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
103	77, 83, 98, 102	mpjao3dan 1307	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> sup ( { A , B } , RR* , < ) < C )
104	14, 103	impbida 596	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) < C <-> ( A < C /\ B < C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 977   /\ w3a 978   = wceq 1353   e. wcel 2148   {cpr 3592   class class class wbr 4000   supcsup 6975   RRcr 7801   +oocpnf 7979   -oocmnf 7980   RR*cxr 7981   < clt 7982   <_ cle 7983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992

This theorem is referenced by:  xrmaxadd  11253  xrltmininf  11262  iooinsup  11269  xmetxpbl  13675  txmetcnp  13685