Theorem sup2 6051

Description: A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent).

Assertion

Ref	Expression
sup2	|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A (y < x \/ y = x)) -> E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z)))

Distinct variable group:   x,y,z,A

Proof of Theorem sup2

Step	Hyp	Ref	Expression
1		peano2re 5436	. . . . . . . . . . . 12 |- (x e. RR -> (x + 1) e. RR)
2	1	adantr 389	. . . . . . . . . . 11 |- ((x e. RR /\ A.y e. A (y < x \/ y = x)) -> (x + 1) e. RR)
3	2	a1i 8	. . . . . . . . . 10 |- (A (_ RR -> ((x e. RR /\ A.y e. A (y < x \/ y = x)) -> (x + 1) e. RR))
4		ssel 2063	. . . . . . . . . . . . . . . . . 18 |- (A (_ RR -> (y e. A -> y e. RR))
5		axlttrn 5504	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((y e. RR /\ x e. RR /\ (x + 1) e. RR) -> ((y < x /\ x < (x + 1)) -> y < (x + 1)))
6	5	3expb 834	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((y e. RR /\ (x e. RR /\ (x + 1) e. RR)) -> ((y < x /\ x < (x + 1)) -> y < (x + 1)))
7	1	ancli 296	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x e. RR -> (x e. RR /\ (x + 1) e. RR))
8	6, 7	sylan2 451	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((y e. RR /\ x e. RR) -> ((y < x /\ x < (x + 1)) -> y < (x + 1)))
9		ltp1t 5811	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x e. RR -> x < (x + 1))
10	8, 9	sylan2i 465	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((y e. RR /\ x e. RR) -> ((y < x /\ x e. RR) -> y < (x + 1)))
11	10	exp4b 379	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. RR -> (x e. RR -> (y < x -> (x e. RR -> y < (x + 1)))))
12	11	com34 36	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (x e. RR -> (x e. RR -> (y < x -> y < (x + 1)))))
13	12	pm2.43d 65	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. RR -> (x e. RR -> (y < x -> y < (x + 1))))
14	13	imp 350	. . . . . . . . . . . . . . . . . . . 20 |- ((y e. RR /\ x e. RR) -> (y < x -> y < (x + 1)))
15		breq1 2622	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = x -> (y < (x + 1) <-> x < (x + 1)))
16	15, 9	syl5cbir 211	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. RR -> (y = x -> y < (x + 1)))
17	16	adantl 388	. . . . . . . . . . . . . . . . . . . 20 |- ((y e. RR /\ x e. RR) -> (y = x -> y < (x + 1)))
18	14, 17	jaod 424	. . . . . . . . . . . . . . . . . . 19 |- ((y e. RR /\ x e. RR) -> ((y < x \/ y = x) -> y < (x + 1)))
19	18	ex 373	. . . . . . . . . . . . . . . . . 18 |- (y e. RR -> (x e. RR -> ((y < x \/ y = x) -> y < (x + 1))))
20	4, 19	syl6 22	. . . . . . . . . . . . . . . . 17 |- (A (_ RR -> (y e. A -> (x e. RR -> ((y < x \/ y = x) -> y < (x + 1)))))
21	20	com23 32	. . . . . . . . . . . . . . . 16 |- (A (_ RR -> (x e. RR -> (y e. A -> ((y < x \/ y = x) -> y < (x + 1)))))
22	21	imp 350	. . . . . . . . . . . . . . 15 |- ((A (_ RR /\ x e. RR) -> (y e. A -> ((y < x \/ y = x) -> y < (x + 1))))
23	22	a2d 13	. . . . . . . . . . . . . 14 |- ((A (_ RR /\ x e. RR) -> ((y e. A -> (y < x \/ y = x)) -> (y e. A -> y < (x + 1))))
24	23	19.20dv 1289	. . . . . . . . . . . . 13 |- ((A (_ RR /\ x e. RR) -> (A.y(y e. A -> (y < x \/ y = x)) -> A.y(y e. A -> y < (x + 1))))
25		df-ral 1649	. . . . . . . . . . . . 13 |- (A.y e. A (y < x \/ y = x) <-> A.y(y e. A -> (y < x \/ y = x)))
26		df-ral 1649	. . . . . . . . . . . . 13 |- (A.y e. A y < (x + 1) <-> A.y(y e. A -> y < (x + 1)))
27	24, 25, 26	3imtr4g 553	. . . . . . . . . . . 12 |- ((A (_ RR /\ x e. RR) -> (A.y e. A (y < x \/ y = x) -> A.y e. A y < (x + 1)))
28	27	ex 373	. . . . . . . . . . 11 |- (A (_ RR -> (x e. RR -> (A.y e. A (y < x \/ y = x) -> A.y e. A y < (x + 1))))
29	28	imp3a 361	. . . . . . . . . 10 |- (A (_ RR -> ((x e. RR /\ A.y e. A (y < x \/ y = x)) -> A.y e. A y < (x + 1)))
30	3, 29	jcad 600	. . . . . . . . 9 |- (A (_ RR -> ((x e. RR /\ A.y e. A (y < x \/ y = x)) -> ((x + 1) e. RR /\ A.y e. A y < (x + 1))))
31		oprex 3983	. . . . . . . . . 10 |- (x + 1) e. V
32		eleq1 1534	. . . . . . . . . . 11 |- (z = (x + 1) -> (z e. RR <-> (x + 1) e. RR))
33		breq2 2623	. . . . . . . . . . . 12 |- (z = (x + 1) -> (y < z <-> y < (x + 1)))
34	33	ralbidv 1663	. . . . . . . . . . 11 |- (z = (x + 1) -> (A.y e. A y < z <-> A.y e. A y < (x + 1)))
35	32, 34	anbi12d 628	. . . . . . . . . 10 |- (z = (x + 1) -> ((z e. RR /\ A.y e. A y < z) <-> ((x + 1) e. RR /\ A.y e. A y < (x + 1))))
36	31, 35	cla4ev 1869	. . . . . . . . 9 |- (((x + 1) e. RR /\ A.y e. A y < (x + 1)) -> E.z(z e. RR /\ A.y e. A y < z))
37	30, 36	syl6 22	. . . . . . . 8 |- (A (_ RR -> ((x e. RR /\ A.y e. A (y < x \/ y = x)) -> E.z(z e. RR /\ A.y e. A y < z)))
38	37	19.23adv 1214	. . . . . . 7 |- (A (_ RR -> (E.x(x e. RR /\ A.y e. A (y < x \/ y = x)) -> E.z(z e. RR /\ A.y e. A y < z)))
39		eleq1 1534	. . . . . . . . 9 |- (z = x -> (z e. RR <-> x e. RR))
40		breq2 2623	. . . . . . . . . 10 |- (z = x -> (y < z <-> y < x))
41	40	ralbidv 1663	. . . . . . . . 9 |- (z = x -> (A.y e. A y < z <-> A.y e. A y < x))
42	39, 41	anbi12d 628	. . . . . . . 8 |- (z = x -> ((z e. RR /\ A.y e. A y < z) <-> (x e. RR /\ A.y e. A y < x)))
43	42	cbvexv 1315	. . . . . . 7 |- (E.z(z e. RR /\ A.y e. A y < z) <-> E.x(x e. RR /\ A.y e. A y < x))
44	38, 43	syl6ib 212	. . . . . 6 |- (A (_ RR -> (E.x(x e. RR /\ A.y e. A (y < x \/ y = x)) -> E.x(x e. RR /\ A.y e. A y < x)))
45		df-rex 1650	. . . . . 6 |- (E.x e. RR A.y e. A (y < x \/ y = x) <-> E.x(x e. RR /\ A.y e. A (y < x \/ y = x)))
46		df-rex 1650	. . . . . 6 |- (E.x e. RR A.y e. A y < x <-> E.x(x e. RR /\ A.y e. A y < x))
47	44, 45, 46	3imtr4g 553	. . . . 5 |- (A (_ RR -> (E.x e. RR A.y e. A (y < x \/ y = x) -> E.x e. RR A.y e. A y < x))
48	47	adantr 389	. . . 4 |- ((A (_ RR /\ A =/= (/)) -> (E.x e. RR A.y e. A (y < x \/ y = x) -> E.x e. RR A.y e. A y < x))
49	48	imdistani 443	. . 3 |- (((A (_ RR /\ A =/= (/)) /\ E.x e. RR A.y e. A (y < x \/ y = x)) -> ((A (_ RR /\ A =/= (/)) /\ E.x e. RR A.y e. A y < x))
50		df-3an 777	. . 3 |- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A (y < x \/ y = x)) <-> ((A (_ RR /\ A =/= (/)) /\ E.x e. RR A.y e. A (y < x \/ y = x)))
51		df-3an 777	. . 3 |- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y < x) <-> ((A (_ RR /\ A =/= (/)) /\ E.x e. RR A.y e. A y < x))
52	49, 50, 51	3imtr4 219	. 2 |- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A (y < x \/ y = x)) -> (A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y < x))
53		axsup 5507	. 2 |- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y < x) -> E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z)))
54	52, 53	syl 10	1 |- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A (y < x \/ y = x)) -> E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  E.wrex 1646   (_ wss 2047