Theorem axpre-sup 9990

Description: A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 10113. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10014. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-sup	|- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <RR x ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) )

Distinct variable group:   x, y, z, A

Proof of Theorem axpre-sup

Dummy variables w v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal2 9953	. . . . . . 7 |- ( x e. RR <-> ( ( 1st ` x ) e. R. /\ x = <. ( 1st ` x ) , 0R >. ) )
2	1	simplbi 476	. . . . . 6 |- ( x e. RR -> ( 1st ` x ) e. R. )
3	2	adantl 482	. . . . 5 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ x e. RR ) -> ( 1st ` x ) e. R. )
4		fo1st 7188	. . . . . . . . . . . 12 |- 1st : _V -onto-> _V
5		fof 6115	. . . . . . . . . . . 12 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
6		ffn 6045	. . . . . . . . . . . 12 |- ( 1st : _V --> _V -> 1st Fn _V )
7	4, 5, 6	mp2b 10	. . . . . . . . . . 11 |- 1st Fn _V
8		ssv 3625	. . . . . . . . . . 11 |- A C_ _V
9		fvelimab 6253	. . . . . . . . . . 11 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( w e. ( 1st " A ) <-> E. y e. A ( 1st ` y ) = w ) )
10	7, 8, 9	mp2an 708	. . . . . . . . . 10 |- ( w e. ( 1st " A ) <-> E. y e. A ( 1st ` y ) = w )
11		r19.29 3072	. . . . . . . . . . . 12 |- ( ( A. y e. A y <RR x /\ E. y e. A ( 1st ` y ) = w ) -> E. y e. A ( y <RR x /\ ( 1st ` y ) = w ) )
12		ssel2 3598	. . . . . . . . . . . . . . . . 17 |- ( ( A C_ RR /\ y e. A ) -> y e. RR )
13		ltresr2 9962	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. RR /\ x e. RR ) -> ( y <RR x <-> ( 1st ` y ) <R ( 1st ` x ) ) )
14		breq1 4656	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1st ` y ) = w -> ( ( 1st ` y ) <R ( 1st ` x ) <-> w <R ( 1st ` x ) ) )
15	13, 14	sylan9bb 736	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( y e. RR /\ x e. RR ) /\ ( 1st ` y ) = w ) -> ( y <RR x <-> w <R ( 1st ` x ) ) )
16	15	biimpd 219	. . . . . . . . . . . . . . . . . 18 |- ( ( ( y e. RR /\ x e. RR ) /\ ( 1st ` y ) = w ) -> ( y <RR x -> w <R ( 1st ` x ) ) )
17	16	exp31 630	. . . . . . . . . . . . . . . . 17 |- ( y e. RR -> ( x e. RR -> ( ( 1st ` y ) = w -> ( y <RR x -> w <R ( 1st ` x ) ) ) ) )
18	12, 17	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( A C_ RR /\ y e. A ) -> ( x e. RR -> ( ( 1st ` y ) = w -> ( y <RR x -> w <R ( 1st ` x ) ) ) ) )
19	18	imp4b 613	. . . . . . . . . . . . . . 15 |- ( ( ( A C_ RR /\ y e. A ) /\ x e. RR ) -> ( ( ( 1st ` y ) = w /\ y <RR x ) -> w <R ( 1st ` x ) ) )
20	19	ancomsd 470	. . . . . . . . . . . . . 14 |- ( ( ( A C_ RR /\ y e. A ) /\ x e. RR ) -> ( ( y <RR x /\ ( 1st ` y ) = w ) -> w <R ( 1st ` x ) ) )
21	20	an32s 846	. . . . . . . . . . . . 13 |- ( ( ( A C_ RR /\ x e. RR ) /\ y e. A ) -> ( ( y <RR x /\ ( 1st ` y ) = w ) -> w <R ( 1st ` x ) ) )
22	21	rexlimdva 3031	. . . . . . . . . . . 12 |- ( ( A C_ RR /\ x e. RR ) -> ( E. y e. A ( y <RR x /\ ( 1st ` y ) = w ) -> w <R ( 1st ` x ) ) )
23	11, 22	syl5 34	. . . . . . . . . . 11 |- ( ( A C_ RR /\ x e. RR ) -> ( ( A. y e. A y <RR x /\ E. y e. A ( 1st ` y ) = w ) -> w <R ( 1st ` x ) ) )
24	23	expd 452	. . . . . . . . . 10 |- ( ( A C_ RR /\ x e. RR ) -> ( A. y e. A y <RR x -> ( E. y e. A ( 1st ` y ) = w -> w <R ( 1st ` x ) ) ) )
25	10, 24	syl7bi 245	. . . . . . . . 9 |- ( ( A C_ RR /\ x e. RR ) -> ( A. y e. A y <RR x -> ( w e. ( 1st " A ) -> w <R ( 1st ` x ) ) ) )
26	25	impr 649	. . . . . . . 8 |- ( ( A C_ RR /\ ( x e. RR /\ A. y e. A y <RR x ) ) -> ( w e. ( 1st " A ) -> w <R ( 1st ` x ) ) )
27	26	adantlr 751	. . . . . . 7 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ ( x e. RR /\ A. y e. A y <RR x ) ) -> ( w e. ( 1st " A ) -> w <R ( 1st ` x ) ) )
28	27	ralrimiv 2965	. . . . . 6 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ ( x e. RR /\ A. y e. A y <RR x ) ) -> A. w e. ( 1st " A ) w <R ( 1st ` x ) )
29	28	expr 643	. . . . 5 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ x e. RR ) -> ( A. y e. A y <RR x -> A. w e. ( 1st " A ) w <R ( 1st ` x ) ) )
30		breq2 4657	. . . . . . 7 |- ( v = ( 1st ` x ) -> ( w <R v <-> w <R ( 1st ` x ) ) )
31	30	ralbidv 2986	. . . . . 6 |- ( v = ( 1st ` x ) -> ( A. w e. ( 1st " A ) w <R v <-> A. w e. ( 1st " A ) w <R ( 1st ` x ) ) )
32	31	rspcev 3309	. . . . 5 |- ( ( ( 1st ` x ) e. R. /\ A. w e. ( 1st " A ) w <R ( 1st ` x ) ) -> E. v e. R. A. w e. ( 1st " A ) w <R v )
33	3, 29, 32	syl6an 568	. . . 4 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ x e. RR ) -> ( A. y e. A y <RR x -> E. v e. R. A. w e. ( 1st " A ) w <R v ) )
34	33	rexlimdva 3031	. . 3 |- ( ( A C_ RR /\ A =/= (/) ) -> ( E. x e. RR A. y e. A y <RR x -> E. v e. R. A. w e. ( 1st " A ) w <R v ) )
35		n0 3931	. . . . . 6 |- ( A =/= (/) <-> E. y y e. A )
36		fnfvima 6496	. . . . . . . . 9 |- ( ( 1st Fn _V /\ A C_ _V /\ y e. A ) -> ( 1st ` y ) e. ( 1st " A ) )
37	7, 8, 36	mp3an12 1414	. . . . . . . 8 |- ( y e. A -> ( 1st ` y ) e. ( 1st " A ) )
38		ne0i 3921	. . . . . . . 8 |- ( ( 1st ` y ) e. ( 1st " A ) -> ( 1st " A ) =/= (/) )
39	37, 38	syl 17	. . . . . . 7 |- ( y e. A -> ( 1st " A ) =/= (/) )
40	39	exlimiv 1858	. . . . . 6 |- ( E. y y e. A -> ( 1st " A ) =/= (/) )
41	35, 40	sylbi 207	. . . . 5 |- ( A =/= (/) -> ( 1st " A ) =/= (/) )
42		supsr 9933	. . . . . 6 |- ( ( ( 1st " A ) =/= (/) /\ E. v e. R. A. w e. ( 1st " A ) w <R v ) -> E. v e. R. ( A. w e. ( 1st " A ) -. v <R w /\ A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) ) )
43	42	ex 450	. . . . 5 |- ( ( 1st " A ) =/= (/) -> ( E. v e. R. A. w e. ( 1st " A ) w <R v -> E. v e. R. ( A. w e. ( 1st " A ) -. v <R w /\ A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) ) ) )
44	41, 43	syl 17	. . . 4 |- ( A =/= (/) -> ( E. v e. R. A. w e. ( 1st " A ) w <R v -> E. v e. R. ( A. w e. ( 1st " A ) -. v <R w /\ A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) ) ) )
45	44	adantl 482	. . 3 |- ( ( A C_ RR /\ A =/= (/) ) -> ( E. v e. R. A. w e. ( 1st " A ) w <R v -> E. v e. R. ( A. w e. ( 1st " A ) -. v <R w /\ A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) ) ) )
46		breq2 4657	. . . . . . . . . . . 12 |- ( w = ( 1st ` y ) -> ( v <R w <-> v <R ( 1st ` y ) ) )
47	46	notbid 308	. . . . . . . . . . 11 |- ( w = ( 1st ` y ) -> ( -. v <R w <-> -. v <R ( 1st ` y ) ) )
48	47	rspccv 3306	. . . . . . . . . 10 |- ( A. w e. ( 1st " A ) -. v <R w -> ( ( 1st ` y ) e. ( 1st " A ) -> -. v <R ( 1st ` y ) ) )
49	37, 48	syl5com 31	. . . . . . . . 9 |- ( y e. A -> ( A. w e. ( 1st " A ) -. v <R w -> -. v <R ( 1st ` y ) ) )
50	49	adantl 482	. . . . . . . 8 |- ( ( A C_ RR /\ y e. A ) -> ( A. w e. ( 1st " A ) -. v <R w -> -. v <R ( 1st ` y ) ) )
51		elreal2 9953	. . . . . . . . . . . . 13 |- ( y e. RR <-> ( ( 1st ` y ) e. R. /\ y = <. ( 1st ` y ) , 0R >. ) )
52	51	simprbi 480	. . . . . . . . . . . 12 |- ( y e. RR -> y = <. ( 1st ` y ) , 0R >. )
53	52	breq2d 4665	. . . . . . . . . . 11 |- ( y e. RR -> ( <. v , 0R >. <RR y <-> <. v , 0R >. <RR <. ( 1st ` y ) , 0R >. ) )
54		ltresr 9961	. . . . . . . . . . 11 |- ( <. v , 0R >. <RR <. ( 1st ` y ) , 0R >. <-> v <R ( 1st ` y ) )
55	53, 54	syl6bb 276	. . . . . . . . . 10 |- ( y e. RR -> ( <. v , 0R >. <RR y <-> v <R ( 1st ` y ) ) )
56	12, 55	syl 17	. . . . . . . . 9 |- ( ( A C_ RR /\ y e. A ) -> ( <. v , 0R >. <RR y <-> v <R ( 1st ` y ) ) )
57	56	notbid 308	. . . . . . . 8 |- ( ( A C_ RR /\ y e. A ) -> ( -. <. v , 0R >. <RR y <-> -. v <R ( 1st ` y ) ) )
58	50, 57	sylibrd 249	. . . . . . 7 |- ( ( A C_ RR /\ y e. A ) -> ( A. w e. ( 1st " A ) -. v <R w -> -. <. v , 0R >. <RR y ) )
59	58	ralrimdva 2969	. . . . . 6 |- ( A C_ RR -> ( A. w e. ( 1st " A ) -. v <R w -> A. y e. A -. <. v , 0R >. <RR y ) )
60	59	ad2antrr 762	. . . . 5 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ v e. R. ) -> ( A. w e. ( 1st " A ) -. v <R w -> A. y e. A -. <. v , 0R >. <RR y ) )
61	52	breq1d 4663	. . . . . . . . . . . . . 14 |- ( y e. RR -> ( y <RR <. v , 0R >. <-> <. ( 1st ` y ) , 0R >. <RR <. v , 0R >. ) )
62		ltresr 9961	. . . . . . . . . . . . . 14 |- ( <. ( 1st ` y ) , 0R >. <RR <. v , 0R >. <-> ( 1st ` y ) <R v )
63	61, 62	syl6bb 276	. . . . . . . . . . . . 13 |- ( y e. RR -> ( y <RR <. v , 0R >. <-> ( 1st ` y ) <R v ) )
64	51	simplbi 476	. . . . . . . . . . . . . . 15 |- ( y e. RR -> ( 1st ` y ) e. R. )
65		breq1 4656	. . . . . . . . . . . . . . . . 17 |- ( w = ( 1st ` y ) -> ( w <R v <-> ( 1st ` y ) <R v ) )
66		breq1 4656	. . . . . . . . . . . . . . . . . 18 |- ( w = ( 1st ` y ) -> ( w <R u <-> ( 1st ` y ) <R u ) )
67	66	rexbidv 3052	. . . . . . . . . . . . . . . . 17 |- ( w = ( 1st ` y ) -> ( E. u e. ( 1st " A ) w <R u <-> E. u e. ( 1st " A ) ( 1st ` y ) <R u ) )
68	65, 67	imbi12d 334	. . . . . . . . . . . . . . . 16 |- ( w = ( 1st ` y ) -> ( ( w <R v -> E. u e. ( 1st " A ) w <R u ) <-> ( ( 1st ` y ) <R v -> E. u e. ( 1st " A ) ( 1st ` y ) <R u ) ) )
69	68	rspccv 3306	. . . . . . . . . . . . . . 15 |- ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> ( ( 1st ` y ) e. R. -> ( ( 1st ` y ) <R v -> E. u e. ( 1st " A ) ( 1st ` y ) <R u ) ) )
70	64, 69	syl5 34	. . . . . . . . . . . . . 14 |- ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> ( y e. RR -> ( ( 1st ` y ) <R v -> E. u e. ( 1st " A ) ( 1st ` y ) <R u ) ) )
71	70	com3l 89	. . . . . . . . . . . . 13 |- ( y e. RR -> ( ( 1st ` y ) <R v -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> E. u e. ( 1st " A ) ( 1st ` y ) <R u ) ) )
72	63, 71	sylbid 230	. . . . . . . . . . . 12 |- ( y e. RR -> ( y <RR <. v , 0R >. -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> E. u e. ( 1st " A ) ( 1st ` y ) <R u ) ) )
73	72	adantr 481	. . . . . . . . . . 11 |- ( ( y e. RR /\ A C_ RR ) -> ( y <RR <. v , 0R >. -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> E. u e. ( 1st " A ) ( 1st ` y ) <R u ) ) )
74		fvelimab 6253	. . . . . . . . . . . . . . . 16 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( u e. ( 1st " A ) <-> E. z e. A ( 1st ` z ) = u ) )
75	7, 8, 74	mp2an 708	. . . . . . . . . . . . . . 15 |- ( u e. ( 1st " A ) <-> E. z e. A ( 1st ` z ) = u )
76		ssel2 3598	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A C_ RR /\ z e. A ) -> z e. RR )
77		ltresr2 9962	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( y e. RR /\ z e. RR ) -> ( y <RR z <-> ( 1st ` y ) <R ( 1st ` z ) ) )
78	76, 77	sylan2 491	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( y e. RR /\ ( A C_ RR /\ z e. A ) ) -> ( y <RR z <-> ( 1st ` y ) <R ( 1st ` z ) ) )
79		breq2 4657	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1st ` z ) = u -> ( ( 1st ` y ) <R ( 1st ` z ) <-> ( 1st ` y ) <R u ) )
80	78, 79	sylan9bb 736	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( y e. RR /\ ( A C_ RR /\ z e. A ) ) /\ ( 1st ` z ) = u ) -> ( y <RR z <-> ( 1st ` y ) <R u ) )
81	80	exbiri 652	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. RR /\ ( A C_ RR /\ z e. A ) ) -> ( ( 1st ` z ) = u -> ( ( 1st ` y ) <R u -> y <RR z ) ) )
82	81	expr 643	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. RR /\ A C_ RR ) -> ( z e. A -> ( ( 1st ` z ) = u -> ( ( 1st ` y ) <R u -> y <RR z ) ) ) )
83	82	com4r 94	. . . . . . . . . . . . . . . . 17 |- ( ( 1st ` y ) <R u -> ( ( y e. RR /\ A C_ RR ) -> ( z e. A -> ( ( 1st ` z ) = u -> y <RR z ) ) ) )
84	83	imp 445	. . . . . . . . . . . . . . . 16 |- ( ( ( 1st ` y ) <R u /\ ( y e. RR /\ A C_ RR ) ) -> ( z e. A -> ( ( 1st ` z ) = u -> y <RR z ) ) )
85	84	reximdvai 3015	. . . . . . . . . . . . . . 15 |- ( ( ( 1st ` y ) <R u /\ ( y e. RR /\ A C_ RR ) ) -> ( E. z e. A ( 1st ` z ) = u -> E. z e. A y <RR z ) )
86	75, 85	syl5bi 232	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` y ) <R u /\ ( y e. RR /\ A C_ RR ) ) -> ( u e. ( 1st " A ) -> E. z e. A y <RR z ) )
87	86	expcom 451	. . . . . . . . . . . . 13 |- ( ( y e. RR /\ A C_ RR ) -> ( ( 1st ` y ) <R u -> ( u e. ( 1st " A ) -> E. z e. A y <RR z ) ) )
88	87	com23 86	. . . . . . . . . . . 12 |- ( ( y e. RR /\ A C_ RR ) -> ( u e. ( 1st " A ) -> ( ( 1st ` y ) <R u -> E. z e. A y <RR z ) ) )
89	88	rexlimdv 3030	. . . . . . . . . . 11 |- ( ( y e. RR /\ A C_ RR ) -> ( E. u e. ( 1st " A ) ( 1st ` y ) <R u -> E. z e. A y <RR z ) )
90	73, 89	syl6d 75	. . . . . . . . . 10 |- ( ( y e. RR /\ A C_ RR ) -> ( y <RR <. v , 0R >. -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> E. z e. A y <RR z ) ) )
91	90	com23 86	. . . . . . . . 9 |- ( ( y e. RR /\ A C_ RR ) -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) )
92	91	ex 450	. . . . . . . 8 |- ( y e. RR -> ( A C_ RR -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) ) )
93	92	com3l 89	. . . . . . 7 |- ( A C_ RR -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> ( y e. RR -> ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) ) )
94	93	ad2antrr 762	. . . . . 6 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ v e. R. ) -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> ( y e. RR -> ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) ) )
95	94	ralrimdv 2968	. . . . 5 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ v e. R. ) -> ( A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) -> A. y e. RR ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) )
96		opelreal 9951	. . . . . . . 8 |- ( <. v , 0R >. e. RR <-> v e. R. )
97	96	biimpri 218	. . . . . . 7 |- ( v e. R. -> <. v , 0R >. e. RR )
98	97	adantl 482	. . . . . 6 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ v e. R. ) -> <. v , 0R >. e. RR )
99		breq1 4656	. . . . . . . . . . 11 |- ( x = <. v , 0R >. -> ( x <RR y <-> <. v , 0R >. <RR y ) )
100	99	notbid 308	. . . . . . . . . 10 |- ( x = <. v , 0R >. -> ( -. x <RR y <-> -. <. v , 0R >. <RR y ) )
101	100	ralbidv 2986	. . . . . . . . 9 |- ( x = <. v , 0R >. -> ( A. y e. A -. x <RR y <-> A. y e. A -. <. v , 0R >. <RR y ) )
102		breq2 4657	. . . . . . . . . . 11 |- ( x = <. v , 0R >. -> ( y <RR x <-> y <RR <. v , 0R >. ) )
103	102	imbi1d 331	. . . . . . . . . 10 |- ( x = <. v , 0R >. -> ( ( y <RR x -> E. z e. A y <RR z ) <-> ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) )
104	103	ralbidv 2986	. . . . . . . . 9 |- ( x = <. v , 0R >. -> ( A. y e. RR ( y <RR x -> E. z e. A y <RR z ) <-> A. y e. RR ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) )
105	101, 104	anbi12d 747	. . . . . . . 8 |- ( x = <. v , 0R >. -> ( ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) <-> ( A. y e. A -. <. v , 0R >. <RR y /\ A. y e. RR ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) ) )
106	105	rspcev 3309	. . . . . . 7 |- ( ( <. v , 0R >. e. RR /\ ( A. y e. A -. <. v , 0R >. <RR y /\ A. y e. RR ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) )
107	106	ex 450	. . . . . 6 |- ( <. v , 0R >. e. RR -> ( ( A. y e. A -. <. v , 0R >. <RR y /\ A. y e. RR ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) )
108	98, 107	syl 17	. . . . 5 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ v e. R. ) -> ( ( A. y e. A -. <. v , 0R >. <RR y /\ A. y e. RR ( y <RR <. v , 0R >. -> E. z e. A y <RR z ) ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) )
109	60, 95, 108	syl2and 500	. . . 4 |- ( ( ( A C_ RR /\ A =/= (/) ) /\ v e. R. ) -> ( ( A. w e. ( 1st " A ) -. v <R w /\ A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) )
110	109	rexlimdva 3031	. . 3 |- ( ( A C_ RR /\ A =/= (/) ) -> ( E. v e. R. ( A. w e. ( 1st " A ) -. v <R w /\ A. w e. R. ( w <R v -> E. u e. ( 1st " A ) w <R u ) ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) )
111	34, 45, 110	3syld 60	. 2 |- ( ( A C_ RR /\ A =/= (/) ) -> ( E. x e. RR A. y e. A y <RR x -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) )
112	111	3impia 1261	1 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <RR x ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   "cima 5117   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   R.cnr 9687   0Rc0r 9688   <R cltr 9693   RRcr 9935   <RR cltrr 9940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-mr 9880  df-ltr 9881  df-0r 9882  df-1r 9883  df-m1r 9884  df-r 9946  df-lt 9949

This theorem is referenced by: (None)