Theorem xrmaxltsup 11968

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 1027	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A e. RR* )
2		simpl2 1028	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B e. RR* )
3		xrmaxcl 11962	. . . . 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 1029	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> C e. RR* )
6		xrmax1sup 11963	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
7	6	3adant3 1044	. . . . 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 10162	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A < C )
11		xrmax2sup 11964	. . . . 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 10162	. . 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 529	. . . . . . 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 536	. . . . . . 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 11965	. . . . . . 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 544	. . . . . . 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 11928	. . . . . . . 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 1274	. . . . . . 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 4136	. . . . 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 529	. . . . . . . . 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 536	. . . . . . . . 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 11920	. . . . . . . . 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 2303	. . . . . . . . 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 10132	. . . . . . 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 4142	. . . . 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 531	. . . . . . 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 10140	. . . . . . . . 9 |- ( A e. RR* -> -. A < -oo )
39	38	3ad2ant1 1045	. . . . . . . 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 4126	. . . . . . 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 680	. . . . . 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 625	. . . . 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 10128	. . . . . . . 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 1047	. . . . . 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 1344	. . . 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 10139	. . . . . . . 8 |- ( C e. RR* -> -. +oo < C )
52	51	3ad2ant3 1047	. . . . . . 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 4124	. . . . . 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 680	. . . . 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 625	. . . 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 10144	. . . . . . . . 9 |- ( B e. RR* -> -oo <_ B )
60	59	3ad2ant2 1046	. . . . . . . 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 4136	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A <_ B )
63		simp1 1024	. . . . . . . 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 1025	. . . . . . . 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 11954	. . . . . . 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 533	. . . . . 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 4136	. . . 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 10128	. . . . . . 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 1045	. . . . 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 1344	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) -> sup ( { A , B } , RR* , < ) < C )
78		simplrr 538	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> B < C )
79		breq1 4117	. . . . . 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 625	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> sup ( { A , B } , RR* , < ) < C )
84		prcom 3772	. . . . . 6 |- { B , A } = { A , B }
85	84	supeq1i 7292	. . . . 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 10144	. . . . . . . . 9 |- ( A e. RR* -> -oo <_ A )
88	87	3ad2ant1 1045	. . . . . . . 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 4136	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B <_ A )
91		simpll2 1064	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B e. RR* )
92		simpll1 1063	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A e. RR* )
93		xrmaxleim 11954	. . . . . . 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 2281	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) = A )
97		simplrl 537	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A < C )
98	96, 97	eqbrtrd 4136	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) < C )
99		elxr 10128	. . . . . 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 1046	. . . 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 1344	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> sup ( { A , B } , RR* , < ) < C )
104	14, 103	impbida 600	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 1004   /\ w3a 1005   = wceq 1398   e. wcel 2205   {cpr 3695   class class class wbr 4114   supcsup 7286   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324   <_ cle 8325

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709

This theorem is referenced by:  xrmaxadd  11971  xrltmininf  11980  iooinsup  11987  xmetxpbl  15499  txmetcnp  15509