Theorem sup3exmid 9230

Description: If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)

Hypothesis

Ref	Expression
sup3exmid.ex	|- ( ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) -> E. x e. RR ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) )

Assertion

Ref	Expression
sup3exmid	|- DECID ph

Distinct variable groups:   x, z   ph, u, w   ph, x, y, z, u

Proof of Theorem sup3exmid

Dummy variables a b j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1 8399	. . . 4 |- 0 < 1
2		0re 8273	. . . . 5 |- 0 e. RR
3		1re 8272	. . . . 5 |- 1 e. RR
4		lttri3 8352	. . . . . . . 8 |- ( ( a e. RR /\ b e. RR ) -> ( a = b <-> ( -. a < b /\ -. b < a ) ) )
5	4	adantl 277	. . . . . . 7 |- ( ( T. /\ ( a e. RR /\ b e. RR ) ) -> ( a = b <-> ( -. a < b /\ -. b < a ) ) )
6		elrabi 2969	. . . . . . . . . . . 12 |- ( k e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> k e. { 0 , 1 } )
7		elpri 3711	. . . . . . . . . . . 12 |- ( k e. { 0 , 1 } -> ( k = 0 \/ k = 1 ) )
8	6, 7	syl 14	. . . . . . . . . . 11 |- ( k e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( k = 0 \/ k = 1 ) )
9		eleq1 2295	. . . . . . . . . . . . 13 |- ( k = 0 -> ( k e. RR <-> 0 e. RR ) )
10	2, 9	mpbiri 168	. . . . . . . . . . . 12 |- ( k = 0 -> k e. RR )
11		eleq1 2295	. . . . . . . . . . . . 13 |- ( k = 1 -> ( k e. RR <-> 1 e. RR ) )
12	3, 11	mpbiri 168	. . . . . . . . . . . 12 |- ( k = 1 -> k e. RR )
13	10, 12	jaoi 724	. . . . . . . . . . 11 |- ( ( k = 0 \/ k = 1 ) -> k e. RR )
14	8, 13	syl 14	. . . . . . . . . 10 |- ( k e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> k e. RR )
15	14	ssriv 3241	. . . . . . . . 9 |- { j e. { 0 , 1 } | ( j = 0 \/ ph ) } C_ RR
16		eqid 2232	. . . . . . . . . . . 12 |- 0 = 0
17	16	orci 739	. . . . . . . . . . 11 |- ( 0 = 0 \/ ph )
18	2	elexi 2825	. . . . . . . . . . . . 13 |- 0 e. _V
19	18	prid1 3796	. . . . . . . . . . . 12 |- 0 e. { 0 , 1 }
20		eqeq1 2239	. . . . . . . . . . . . . 14 |- ( j = 0 -> ( j = 0 <-> 0 = 0 ) )
21	20	orbi1d 799	. . . . . . . . . . . . 13 |- ( j = 0 -> ( ( j = 0 \/ ph ) <-> ( 0 = 0 \/ ph ) ) )
22	21	elrab3 2973	. . . . . . . . . . . 12 |- ( 0 e. { 0 , 1 } -> ( 0 e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } <-> ( 0 = 0 \/ ph ) ) )
23	19, 22	ax-mp 5	. . . . . . . . . . 11 |- ( 0 e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } <-> ( 0 = 0 \/ ph ) )
24	17, 23	mpbir 146	. . . . . . . . . 10 |- 0 e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) }
25		elex2 2829	. . . . . . . . . 10 |- ( 0 e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> E. w w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } )
26	24, 25	ax-mp 5	. . . . . . . . 9 |- E. w w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) }
27		elrabi 2969	. . . . . . . . . . . 12 |- ( y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> y e. { 0 , 1 } )
28		elpri 3711	. . . . . . . . . . . . 13 |- ( y e. { 0 , 1 } -> ( y = 0 \/ y = 1 ) )
29		0le1 8754	. . . . . . . . . . . . . . 15 |- 0 <_ 1
30		breq1 4111	. . . . . . . . . . . . . . 15 |- ( y = 0 -> ( y <_ 1 <-> 0 <_ 1 ) )
31	29, 30	mpbiri 168	. . . . . . . . . . . . . 14 |- ( y = 0 -> y <_ 1 )
32	3	eqlei2 8367	. . . . . . . . . . . . . 14 |- ( y = 1 -> y <_ 1 )
33	31, 32	jaoi 724	. . . . . . . . . . . . 13 |- ( ( y = 0 \/ y = 1 ) -> y <_ 1 )
34	28, 33	syl 14	. . . . . . . . . . . 12 |- ( y e. { 0 , 1 } -> y <_ 1 )
35	27, 34	syl 14	. . . . . . . . . . 11 |- ( y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> y <_ 1 )
36	35	rgen 2595	. . . . . . . . . 10 |- A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ 1
37		breq2 4112	. . . . . . . . . . . 12 |- ( x = 1 -> ( y <_ x <-> y <_ 1 ) )
38	37	ralbidv 2542	. . . . . . . . . . 11 |- ( x = 1 -> ( A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x <-> A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ 1 ) )
39	38	rspcev 2920	. . . . . . . . . 10 |- ( ( 1 e. RR /\ A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ 1 ) -> E. x e. RR A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x )
40	3, 36, 39	mp2an 426	. . . . . . . . 9 |- E. x e. RR A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x
41		prexg 4324	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ 1 e. RR ) -> { 0 , 1 } e. _V )
42	2, 3, 41	mp2an 426	. . . . . . . . . . 11 |- { 0 , 1 } e. _V
43	42	rabex 4255	. . . . . . . . . 10 |- { j e. { 0 , 1 } | ( j = 0 \/ ph ) } e. _V
44		sseq1 3260	. . . . . . . . . . . 12 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( u C_ RR <-> { j e. { 0 , 1 } | ( j = 0 \/ ph ) } C_ RR ) )
45		eleq2 2296	. . . . . . . . . . . . 13 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( w e. u <-> w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } ) )
46	45	exbidv 1874	. . . . . . . . . . . 12 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( E. w w e. u <-> E. w w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } ) )
47		raleq 2740	. . . . . . . . . . . . 13 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( A. y e. u y <_ x <-> A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x ) )
48	47	rexbidv 2543	. . . . . . . . . . . 12 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( E. x e. RR A. y e. u y <_ x <-> E. x e. RR A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x ) )
49	44, 46, 48	3anbi123d 1349	. . . . . . . . . . 11 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) <-> ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } C_ RR /\ E. w w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } /\ E. x e. RR A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x ) ) )
50		raleq 2740	. . . . . . . . . . . . 13 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( A. y e. u -. x < y <-> A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -. x < y ) )
51		rexeq 2741	. . . . . . . . . . . . . . 15 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( E. z e. u y < z <-> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) )
52	51	imbi2d 230	. . . . . . . . . . . . . 14 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( ( y < x -> E. z e. u y < z ) <-> ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) ) )
53	52	ralbidv 2542	. . . . . . . . . . . . 13 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( A. y e. RR ( y < x -> E. z e. u y < z ) <-> A. y e. RR ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) ) )
54	50, 53	anbi12d 473	. . . . . . . . . . . 12 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) <-> ( A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) ) ) )
55	54	rexbidv 2543	. . . . . . . . . . 11 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( E. x e. RR ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) <-> E. x e. RR ( A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) ) ) )
56	49, 55	imbi12d 234	. . . . . . . . . 10 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( ( ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) -> E. x e. RR ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) ) <-> ( ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } C_ RR /\ E. w w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } /\ E. x e. RR A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x ) -> E. x e. RR ( A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) ) ) ) )
57		sup3exmid.ex	. . . . . . . . . 10 |- ( ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) -> E. x e. RR ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) )
58	43, 56, 57	vtocl 2868	. . . . . . . . 9 |- ( ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } C_ RR /\ E. w w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } /\ E. x e. RR A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ x ) -> E. x e. RR ( A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) ) )
59	15, 26, 40, 58	mp3an 1374	. . . . . . . 8 |- E. x e. RR ( A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) )
60	59	a1i 9	. . . . . . 7 |- ( T. -> E. x e. RR ( A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y < z ) ) )
61	5, 60	supclti 7288	. . . . . 6 |- ( T. -> sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) e. RR )
62	61	mptru 1407	. . . . 5 |- sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) e. RR
63		axltwlin 8340	. . . . 5 |- ( ( 0 e. RR /\ 1 e. RR /\ sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) e. RR ) -> ( 0 < 1 -> ( 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) \/ sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 ) ) )
64	2, 3, 62, 63	mp3an 1374	. . . 4 |- ( 0 < 1 -> ( 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) \/ sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 ) )
65	1, 64	ax-mp 5	. . 3 |- ( 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) \/ sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 )
66	5, 60	suplubti 7290	. . . . . . . 8 |- ( T. -> ( ( 0 e. RR /\ 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) ) -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } 0 < z ) )
67	66	mptru 1407	. . . . . . 7 |- ( ( 0 e. RR /\ 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) ) -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } 0 < z )
68	2, 67	mpan 424	. . . . . 6 |- ( 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) -> E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } 0 < z )
69		df-rex 2526	. . . . . 6 |- ( E. z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } 0 < z <-> E. z ( z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } /\ 0 < z ) )
70	68, 69	sylib 122	. . . . 5 |- ( 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) -> E. z ( z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } /\ 0 < z ) )
71		eqeq1 2239	. . . . . . . . 9 |- ( j = z -> ( j = 0 <-> z = 0 ) )
72	71	orbi1d 799	. . . . . . . 8 |- ( j = z -> ( ( j = 0 \/ ph ) <-> ( z = 0 \/ ph ) ) )
73	72	elrab 2972	. . . . . . 7 |- ( z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } <-> ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) )
74		simpr 110	. . . . . . . . . 10 |- ( ( ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) /\ 0 < z ) -> 0 < z )
75	74	gt0ne0d 8785	. . . . . . . . 9 |- ( ( ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) /\ 0 < z ) -> z =/= 0 )
76	75	neneqd 2433	. . . . . . . 8 |- ( ( ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) /\ 0 < z ) -> -. z = 0 )
77		simplr 529	. . . . . . . 8 |- ( ( ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) /\ 0 < z ) -> ( z = 0 \/ ph ) )
78		orel1 733	. . . . . . . 8 |- ( -. z = 0 -> ( ( z = 0 \/ ph ) -> ph ) )
79	76, 77, 78	sylc 62	. . . . . . 7 |- ( ( ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) /\ 0 < z ) -> ph )
80	73, 79	sylanb 284	. . . . . 6 |- ( ( z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } /\ 0 < z ) -> ph )
81	80	exlimiv 1647	. . . . 5 |- ( E. z ( z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } /\ 0 < z ) -> ph )
82	70, 81	syl 14	. . . 4 |- ( 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) -> ph )
83	3	ltnri 8365	. . . . . 6 |- -. 1 < 1
84		iba 300	. . . . . . . . . . . 12 |- ( ( z = 0 \/ ph ) -> ( z e. { 0 , 1 } <-> ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) ) )
85	84	olcs 744	. . . . . . . . . . 11 |- ( ph -> ( z e. { 0 , 1 } <-> ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) ) )
86	73, 85	bitr4id 199	. . . . . . . . . 10 |- ( ph -> ( z e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } <-> z e. { 0 , 1 } ) )
87	86	eqrdv 2230	. . . . . . . . 9 |- ( ph -> { j e. { 0 , 1 } | ( j = 0 \/ ph ) } = { 0 , 1 } )
88	87	supeq1d 7277	. . . . . . . 8 |- ( ph -> sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) = sup ( { 0 , 1 } , RR , < ) )
89	3	a1i 9	. . . . . . . . . 10 |- ( T. -> 1 e. RR )
90	3	elexi 2825	. . . . . . . . . . . 12 |- 1 e. _V
91	90	prid2 3797	. . . . . . . . . . 11 |- 1 e. { 0 , 1 }
92	91	a1i 9	. . . . . . . . . 10 |- ( T. -> 1 e. { 0 , 1 } )
93		elpri 3711	. . . . . . . . . . . 12 |- ( z e. { 0 , 1 } -> ( z = 0 \/ z = 1 ) )
94	2, 3	lenlti 8373	. . . . . . . . . . . . . . 15 |- ( 0 <_ 1 <-> -. 1 < 0 )
95	29, 94	mpbi 145	. . . . . . . . . . . . . 14 |- -. 1 < 0
96		breq2 4112	. . . . . . . . . . . . . 14 |- ( z = 0 -> ( 1 < z <-> 1 < 0 ) )
97	95, 96	mtbiri 682	. . . . . . . . . . . . 13 |- ( z = 0 -> -. 1 < z )
98		breq2 4112	. . . . . . . . . . . . . 14 |- ( z = 1 -> ( 1 < z <-> 1 < 1 ) )
99	83, 98	mtbiri 682	. . . . . . . . . . . . 13 |- ( z = 1 -> -. 1 < z )
100	97, 99	jaoi 724	. . . . . . . . . . . 12 |- ( ( z = 0 \/ z = 1 ) -> -. 1 < z )
101	93, 100	syl 14	. . . . . . . . . . 11 |- ( z e. { 0 , 1 } -> -. 1 < z )
102	101	adantl 277	. . . . . . . . . 10 |- ( ( T. /\ z e. { 0 , 1 } ) -> -. 1 < z )
103	5, 89, 92, 102	supmaxti 7294	. . . . . . . . 9 |- ( T. -> sup ( { 0 , 1 } , RR , < ) = 1 )
104	103	mptru 1407	. . . . . . . 8 |- sup ( { 0 , 1 } , RR , < ) = 1
105	88, 104	eqtrdi 2281	. . . . . . 7 |- ( ph -> sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) = 1 )
106	105	breq1d 4118	. . . . . 6 |- ( ph -> ( sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 <-> 1 < 1 ) )
107	83, 106	mtbiri 682	. . . . 5 |- ( ph -> -. sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 )
108	107	con2i 632	. . . 4 |- ( sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 -> -. ph )
109	82, 108	orim12i 767	. . 3 |- ( ( 0 < sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) \/ sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 ) -> ( ph \/ -. ph ) )
110	65, 109	ax-mp 5	. 2 |- ( ph \/ -. ph )
111		df-dc 843	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
112	110, 111	mpbir 146	1 |- DECID ph

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   T. wtru 1399   E.wex 1541   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   _Vcvv 2812   C_ wss 3210   {cpr 3689   class class class wbr 4108   supcsup 7272   RRcr 8125   0cc0 8126   1c1 8127   < clt 8307   <_ cle 8308

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223  ax-0lt1 8232  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-xp 4754  df-cnv 4756  df-iota 5311  df-riota 6002  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313

This theorem is referenced by: (None)