Theorem xrmaxltsup 11440

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 1002	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A e. RR* )
2		simpl2 1003	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B e. RR* )
3		xrmaxcl 11434	. . . . 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 1004	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> C e. RR* )
6		xrmax1sup 11435	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
7	6	3adant3 1019	. . . . 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 9902	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A < C )
11		xrmax2sup 11436	. . . . 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 9902	. . 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 11437	. . . . . . 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 11400	. . . . . . . 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 1249	. . . . . . 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 4056	. . . . 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 11392	. . . . . . . . 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 2265	. . . . . . . . 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 9872	. . . . . . 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 4062	. . . . 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 9880	. . . . . . . . 9 |- ( A e. RR* -> -. A < -oo )
39	38	3ad2ant1 1020	. . . . . . . 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 4046	. . . . . . 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 674	. . . . . 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 9868	. . . . . . . 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 1022	. . . . . 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 1318	. . . 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 9879	. . . . . . . 8 |- ( C e. RR* -> -. +oo < C )
52	51	3ad2ant3 1022	. . . . . . 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 4044	. . . . . 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 674	. . . . 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 9884	. . . . . . . . 9 |- ( B e. RR* -> -oo <_ B )
60	59	3ad2ant2 1021	. . . . . . . 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 4056	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A <_ B )
63		simp1 999	. . . . . . . 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 1000	. . . . . . . 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 11426	. . . . . . 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 4056	. . . 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 9868	. . . . . . 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 1020	. . . . 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 1318	. . 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 4037	. . . . . 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 3699	. . . . . 6 |- { B , A } = { A , B }
85	84	supeq1i 7063	. . . . 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 9884	. . . . . . . . 9 |- ( A e. RR* -> -oo <_ A )
88	87	3ad2ant1 1020	. . . . . . . 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 4056	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B <_ A )
91		simpll2 1039	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B e. RR* )
92		simpll1 1038	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A e. RR* )
93		xrmaxleim 11426	. . . . . . 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 2243	. . . 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 4056	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) < C )
99		elxr 9868	. . . . . 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 1021	. . . 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 1318	. 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 979   /\ w3a 980   = wceq 1364   e. wcel 2167   {cpr 3624   class class class wbr 4034   supcsup 7057   RRcr 7895   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077   < clt 8078   <_ cle 8079

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-sup 7059  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181

This theorem is referenced by:  xrmaxadd  11443  xrltmininf  11452  iooinsup  11459  xmetxpbl  14828  txmetcnp  14838