Theorem sup3exmid 9137

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 8306	. . . 4 |- 0 < 1
2		0re 8179	. . . . 5 |- 0 e. RR
3		1re 8178	. . . . 5 |- 1 e. RR
4		lttri3 8259	. . . . . . . 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 2959	. . . . . . . . . . . 12 |- ( k e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> k e. { 0 , 1 } )
7		elpri 3692	. . . . . . . . . . . 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 2294	. . . . . . . . . . . . 13 |- ( k = 0 -> ( k e. RR <-> 0 e. RR ) )
10	2, 9	mpbiri 168	. . . . . . . . . . . 12 |- ( k = 0 -> k e. RR )
11		eleq1 2294	. . . . . . . . . . . . 13 |- ( k = 1 -> ( k e. RR <-> 1 e. RR ) )
12	3, 11	mpbiri 168	. . . . . . . . . . . 12 |- ( k = 1 -> k e. RR )
13	10, 12	jaoi 723	. . . . . . . . . . 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 3231	. . . . . . . . 9 |- { j e. { 0 , 1 } | ( j = 0 \/ ph ) } C_ RR
16		eqid 2231	. . . . . . . . . . . 12 |- 0 = 0
17	16	orci 738	. . . . . . . . . . 11 |- ( 0 = 0 \/ ph )
18	2	elexi 2815	. . . . . . . . . . . . 13 |- 0 e. _V
19	18	prid1 3777	. . . . . . . . . . . 12 |- 0 e. { 0 , 1 }
20		eqeq1 2238	. . . . . . . . . . . . . 14 |- ( j = 0 -> ( j = 0 <-> 0 = 0 ) )
21	20	orbi1d 798	. . . . . . . . . . . . 13 |- ( j = 0 -> ( ( j = 0 \/ ph ) <-> ( 0 = 0 \/ ph ) ) )
22	21	elrab3 2963	. . . . . . . . . . . 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 2819	. . . . . . . . . 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 2959	. . . . . . . . . . . 12 |- ( y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> y e. { 0 , 1 } )
28		elpri 3692	. . . . . . . . . . . . 13 |- ( y e. { 0 , 1 } -> ( y = 0 \/ y = 1 ) )
29		0le1 8661	. . . . . . . . . . . . . . 15 |- 0 <_ 1
30		breq1 4091	. . . . . . . . . . . . . . 15 |- ( y = 0 -> ( y <_ 1 <-> 0 <_ 1 ) )
31	29, 30	mpbiri 168	. . . . . . . . . . . . . 14 |- ( y = 0 -> y <_ 1 )
32	3	eqlei2 8274	. . . . . . . . . . . . . 14 |- ( y = 1 -> y <_ 1 )
33	31, 32	jaoi 723	. . . . . . . . . . . . 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 2585	. . . . . . . . . 10 |- A. y e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } y <_ 1
37		breq2 4092	. . . . . . . . . . . 12 |- ( x = 1 -> ( y <_ x <-> y <_ 1 ) )
38	37	ralbidv 2532	. . . . . . . . . . 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 2910	. . . . . . . . . 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 4301	. . . . . . . . . . . 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 4234	. . . . . . . . . 10 |- { j e. { 0 , 1 } | ( j = 0 \/ ph ) } e. _V
44		sseq1 3250	. . . . . . . . . . . 12 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( u C_ RR <-> { j e. { 0 , 1 } | ( j = 0 \/ ph ) } C_ RR ) )
45		eleq2 2295	. . . . . . . . . . . . 13 |- ( u = { j e. { 0 , 1 } | ( j = 0 \/ ph ) } -> ( w e. u <-> w e. { j e. { 0 , 1 } | ( j = 0 \/ ph ) } ) )
46	45	exbidv 1873	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . 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 2533	. . . . . . . . . . . 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 1348	. . . . . . . . . . 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 2730	. . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . 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 2532	. . . . . . . . . . . . 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 2533	. . . . . . . . . . 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 2858	. . . . . . . . 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 1373	. . . . . . . 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 7197	. . . . . 6 |- ( T. -> sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) e. RR )
62	61	mptru 1406	. . . . 5 |- sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) e. RR
63		axltwlin 8247	. . . . 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 1373	. . . 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 7199	. . . . . . . 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 1406	. . . . . . 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 2516	. . . . . 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 2238	. . . . . . . . 9 |- ( j = z -> ( j = 0 <-> z = 0 ) )
72	71	orbi1d 798	. . . . . . . 8 |- ( j = z -> ( ( j = 0 \/ ph ) <-> ( z = 0 \/ ph ) ) )
73	72	elrab 2962	. . . . . . 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 8692	. . . . . . . . 9 |- ( ( ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) /\ 0 < z ) -> z =/= 0 )
76	75	neneqd 2423	. . . . . . . 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 732	. . . . . . . 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 1646	. . . . 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 8272	. . . . . 6 |- -. 1 < 1
84		iba 300	. . . . . . . . . . . 12 |- ( ( z = 0 \/ ph ) -> ( z e. { 0 , 1 } <-> ( z e. { 0 , 1 } /\ ( z = 0 \/ ph ) ) ) )
85	84	olcs 743	. . . . . . . . . . 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 2229	. . . . . . . . 9 |- ( ph -> { j e. { 0 , 1 } | ( j = 0 \/ ph ) } = { 0 , 1 } )
88	87	supeq1d 7186	. . . . . . . 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 2815	. . . . . . . . . . . 12 |- 1 e. _V
91	90	prid2 3778	. . . . . . . . . . 11 |- 1 e. { 0 , 1 }
92	91	a1i 9	. . . . . . . . . 10 |- ( T. -> 1 e. { 0 , 1 } )
93		elpri 3692	. . . . . . . . . . . 12 |- ( z e. { 0 , 1 } -> ( z = 0 \/ z = 1 ) )
94	2, 3	lenlti 8280	. . . . . . . . . . . . . . 15 |- ( 0 <_ 1 <-> -. 1 < 0 )
95	29, 94	mpbi 145	. . . . . . . . . . . . . 14 |- -. 1 < 0
96		breq2 4092	. . . . . . . . . . . . . 14 |- ( z = 0 -> ( 1 < z <-> 1 < 0 ) )
97	95, 96	mtbiri 681	. . . . . . . . . . . . 13 |- ( z = 0 -> -. 1 < z )
98		breq2 4092	. . . . . . . . . . . . . 14 |- ( z = 1 -> ( 1 < z <-> 1 < 1 ) )
99	83, 98	mtbiri 681	. . . . . . . . . . . . 13 |- ( z = 1 -> -. 1 < z )
100	97, 99	jaoi 723	. . . . . . . . . . . 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 7203	. . . . . . . . 9 |- ( T. -> sup ( { 0 , 1 } , RR , < ) = 1 )
104	103	mptru 1406	. . . . . . . 8 |- sup ( { 0 , 1 } , RR , < ) = 1
105	88, 104	eqtrdi 2280	. . . . . . 7 |- ( ph -> sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) = 1 )
106	105	breq1d 4098	. . . . . 6 |- ( ph -> ( sup ( { j e. { 0 , 1 } | ( j = 0 \/ ph ) } , RR , < ) < 1 <-> 1 < 1 ) )
107	83, 106	mtbiri 681	. . . . 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 766	. . 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 842	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
112	110, 111	mpbir 146	1 |- DECID ph

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   T. wtru 1398   E.wex 1540   e. wcel 2202   A.wral 2510   E.wrex 2511   {crab 2514   _Vcvv 2802   C_ wss 3200   {cpr 3670   class class class wbr 4088   supcsup 7181   RRcr 8031   0cc0 8032   1c1 8033   < clt 8214   <_ cle 8215

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129  ax-0lt1 8138  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-iota 5286  df-riota 5971  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220

This theorem is referenced by: (None)