Theorem xrmaxltsup 11644

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 1003	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A e. RR* )
2		simpl2 1004	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B e. RR* )
3		xrmaxcl 11638	. . . . 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 1005	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> C e. RR* )
6		xrmax1sup 11639	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
7	6	3adant3 1020	. . . . 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 9952	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A < C )
11		xrmax2sup 11640	. . . . 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 9952	. . 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 11641	. . . . . . 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 11604	. . . . . . . 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 1250	. . . . . . 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 4073	. . . . 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 11596	. . . . . . . . 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 2275	. . . . . . . . 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 9922	. . . . . . 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 4079	. . . . 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 9930	. . . . . . . . 9 |- ( A e. RR* -> -. A < -oo )
39	38	3ad2ant1 1021	. . . . . . . 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 4063	. . . . . . 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 675	. . . . . 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 621	. . . . 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 9918	. . . . . . . 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 1023	. . . . . 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 1320	. . . 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 9929	. . . . . . . 8 |- ( C e. RR* -> -. +oo < C )
52	51	3ad2ant3 1023	. . . . . . 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 4061	. . . . . 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 675	. . . . 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 621	. . . 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 9934	. . . . . . . . 9 |- ( B e. RR* -> -oo <_ B )
60	59	3ad2ant2 1022	. . . . . . . 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 4073	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A <_ B )
63		simp1 1000	. . . . . . . 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 1001	. . . . . . . 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 11630	. . . . . . 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 4073	. . . 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 9918	. . . . . . 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 1021	. . . . 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 1320	. . 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 4054	. . . . . 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 621	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> sup ( { A , B } , RR* , < ) < C )
84		prcom 3714	. . . . . 6 |- { B , A } = { A , B }
85	84	supeq1i 7105	. . . . 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 9934	. . . . . . . . 9 |- ( A e. RR* -> -oo <_ A )
88	87	3ad2ant1 1021	. . . . . . . 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 4073	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B <_ A )
91		simpll2 1040	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B e. RR* )
92		simpll1 1039	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A e. RR* )
93		xrmaxleim 11630	. . . . . . 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 2253	. . . 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 4073	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) < C )
99		elxr 9918	. . . . . 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 1022	. . . 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 1320	. 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 980   /\ w3a 981   = wceq 1373   e. wcel 2177   {cpr 3639   class class class wbr 4051   supcsup 7099   RRcr 7944   +oocpnf 8124   -oocmnf 8125   RR*cxr 8126   < clt 8127   <_ cle 8128

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-sup 7101  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-rp 9796  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385

This theorem is referenced by:  xrmaxadd  11647  xrltmininf  11656  iooinsup  11663  xmetxpbl  15055  txmetcnp  15065