Theorem xrmaxltsup 11027

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 984	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A e. RR* )
2		simpl2 985	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B e. RR* )
3		xrmaxcl 11021	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> sup ( { A , B } , RR* , < ) e. RR* )
4	1, 2, 3	syl2anc 408	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> sup ( { A , B } , RR* , < ) e. RR* )
5		simpl3 986	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> C e. RR* )
6		xrmax1sup 11022	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
7	6	3adant3 1001	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
8	7	adantr 274	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A <_ sup ( { A , B } , RR* , < ) )
9		simpr 109	. . . 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 9593	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> A < C )
11		xrmax2sup 11023	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ sup ( { A , B } , RR* , < ) )
12	1, 2, 11	syl2anc 408	. . . 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 9593	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> B < C )
14	10, 13	jca 304	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ sup ( { A , B } , RR* , < ) < C ) -> ( A < C /\ B < C ) )
15		simplr 519	. . . . . . 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 523	. . . . . . 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 11024	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR* , < ) = sup ( { A , B } , RR , < ) )
18	15, 16, 17	syl2anc 408	. . . . . 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 531	. . . . . . 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 109	. . . . . . . 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 10990	. . . . . . . 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 1216	. . . . . . 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 166	. . . . . 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 3950	. . . . 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 519	. . . . . . . . 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 523	. . . . . . . . 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 10982	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
28	25, 26, 27	syl2anc 408	. . . . . . . 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 2208	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( { A , B } , RR* , < ) e. RR <-> sup ( { A , B } , RR , < ) e. RR ) )
30	25, 26, 29	syl2anc 408	. . . . . . . 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 166	. . . . . . 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 9567	. . . . . . 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 109	. . . . . 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 3956	. . . . 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 520	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> A < C )
37	36	ad3antrrr 483	. . . . . 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 9574	. . . . . . . . 9 |- ( A e. RR* -> -. A < -oo )
39	38	3ad2ant1 1002	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. A < -oo )
40	39	ad4antr 485	. . . . . . 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 109	. . . . . . . 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 3941	. . . . . . 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 662	. . . . . 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 609	. . . . 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 9563	. . . . . . . 8 |- ( C e. RR* <-> ( C e. RR \/ C = +oo \/ C = -oo ) )
46	45	biimpi 119	. . . . . . 7 |- ( C e. RR* -> ( C e. RR \/ C = +oo \/ C = -oo ) )
47	46	3ad2ant3 1004	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. RR \/ C = +oo \/ C = -oo ) )
48	47	ad3antrrr 483	. . . . 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 1285	. . . 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 479	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> A < C )
51		pnfnlt 9573	. . . . . . . 8 |- ( C e. RR* -> -. +oo < C )
52	51	3ad2ant3 1004	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. +oo < C )
53	52	ad3antrrr 483	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> -. +oo < C )
54		simpr 109	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = +oo ) -> A = +oo )
55	54	breq1d 3939	. . . . . 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 662	. . . . 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 609	. . . 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 109	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A = -oo )
59		mnfle 9578	. . . . . . . . 9 |- ( B e. RR* -> -oo <_ B )
60	59	3ad2ant2 1003	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -oo <_ B )
61	60	ad3antrrr 483	. . . . . . 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 3950	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A <_ B )
63		simp1 981	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
64	63	ad3antrrr 483	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> A e. RR* )
65		simp2 982	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
66	65	ad3antrrr 483	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) /\ A = -oo ) -> B e. RR* )
67		xrmaxleim 11013	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )
68	64, 66, 67	syl2anc 408	. . . . . 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 521	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> B < C )
71	70	ad2antrr 479	. . . . 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 3950	. . . 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 9563	. . . . . . 7 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
74	73	biimpi 119	. . . . . 6 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
75	74	3ad2ant1 1002	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
76	75	ad2antrr 479	. . . 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 1285	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B e. RR ) -> sup ( { A , B } , RR* , < ) < C )
78		simplrr 525	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> B < C )
79		breq1 3932	. . . . . 6 |- ( B = +oo -> ( B < C <-> +oo < C ) )
80	79	adantl 275	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> ( B < C <-> +oo < C ) )
81	78, 80	mpbid 146	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> +oo < C )
82	52	ad2antrr 479	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> -. +oo < C )
83	81, 82	pm2.21dd 609	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = +oo ) -> sup ( { A , B } , RR* , < ) < C )
84		prcom 3599	. . . . . 6 |- { B , A } = { A , B }
85	84	supeq1i 6875	. . . . 5 |- sup ( { B , A } , RR* , < ) = sup ( { A , B } , RR* , < )
86		simpr 109	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B = -oo )
87		mnfle 9578	. . . . . . . . 9 |- ( A e. RR* -> -oo <_ A )
88	87	3ad2ant1 1002	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -oo <_ A )
89	88	ad2antrr 479	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> -oo <_ A )
90	86, 89	eqbrtrd 3950	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B <_ A )
91		simpll2 1021	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> B e. RR* )
92		simpll1 1020	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A e. RR* )
93		xrmaxleim 11013	. . . . . . 7 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A -> sup ( { B , A } , RR* , < ) = A ) )
94	91, 92, 93	syl2anc 408	. . . . . 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	syl5eqr 2186	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) = A )
97		simplrl 524	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> A < C )
98	96, 97	eqbrtrd 3950	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) /\ B = -oo ) -> sup ( { A , B } , RR* , < ) < C )
99		elxr 9563	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
100	99	biimpi 119	. . . . 5 |- ( B e. RR* -> ( B e. RR \/ B = +oo \/ B = -oo ) )
101	100	3ad2ant2 1003	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
102	101	adantr 274	. . 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 1285	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ B < C ) ) -> sup ( { A , B } , RR* , < ) < C )
104	14, 103	impbida 585	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 103   <-> wb 104   \/ w3o 961   /\ w3a 962   = wceq 1331   e. wcel 1480   {cpr 3528   class class class wbr 3929   supcsup 6869   RRcr 7619   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   < clt 7800   <_ cle 7801

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771

This theorem is referenced by:  xrmaxadd  11030  xrltmininf  11039  iooinsup  11046  xmetxpbl  12677  txmetcnp  12687