Theorem sup2 6053

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 5448	. . . . . . . . . . . 12 |- (x e. RR -> (x + 1) e. RR)
2	1	adantr 391	. . . . . . . . . . 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 2066	. . . . . . . . . . . . . . . . 17 |- (A (_ RR -> (y e. A -> y e. RR))
5		axlttrn 5516	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((y e. RR /\ x e. RR /\ (x + 1) e. RR) -> ((y < x /\ x < (x + 1)) -> y < (x + 1)))
6	5	3expb 836	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((y e. RR /\ (x e. RR /\ (x + 1) e. RR)) -> ((y < x /\ x < (x + 1)) -> y < (x + 1)))
7	1	ancli 296	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x e. RR -> (x e. RR /\ (x + 1) e. RR))
8	6, 7	sylan2 453	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((y e. RR /\ x e. RR) -> ((y < x /\ x < (x + 1)) -> y < (x + 1)))
9		ltp1t 5813	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x e. RR -> x < (x + 1))
10	8, 9	sylan2i 467	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y e. RR /\ x e. RR) -> ((y < x /\ x e. RR) -> y < (x + 1)))
11	10	exp4b 381	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (x e. RR -> (y < x -> (x e. RR -> y < (x + 1)))))
12	11	com34 36	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. RR -> (x e. RR -> (x e. RR -> (y < x -> y < (x + 1)))))
13	12	pm2.43d 65	. . . . . . . . . . . . . . . . . . . 20 |- (y e. RR -> (x e. RR -> (y < x -> y < (x + 1))))
14	13	imp 350	. . . . . . . . . . . . . . . . . . 19 |- ((y e. RR /\ x e. RR) -> (y < x -> y < (x + 1)))
15		breq1 2627	. . . . . . . . . . . . . . . . . . . . 21 |- (y = x -> (y < (x + 1) <-> x < (x + 1)))
16	15, 9	syl5cbir 211	. . . . . . . . . . . . . . . . . . . 20 |- (x e. RR -> (y = x -> y < (x + 1)))
17	16	adantl 390	. . . . . . . . . . . . . . . . . . 19 |- ((y e. RR /\ x e. RR) -> (y = x -> y < (x + 1)))
18	14, 17	jaod 426	. . . . . . . . . . . . . . . . . 18 |- ((y e. RR /\ x e. RR) -> ((y < x \/ y = x) -> y < (x + 1)))
19	18	ex 373	. . . . . . . . . . . . . . . . 17 |- (y e. RR -> (x e. RR -> ((y < x \/ y = x) -> y < (x + 1))))
20	4, 19	syl6 22	. . . . . . . . . . . . . . . 16 |- (A (_ RR -> (y e. A -> (x e. RR -> ((y < x \/ y = x) -> y < (x + 1)))))
21	20	com23 32	. . . . . . . . . . . . . . 15 |- (A (_ RR -> (x e. RR -> (y e. A -> ((y < x \/ y = x) -> y < (x + 1)))))
22	21	imp 350	. . . . . . . . . . . . . 14 |- ((A (_ RR /\ x e. RR) -> (y e. A -> ((y < x \/ y = x) -> y < (x + 1))))
23	22	a2d 13	. . . . . . . . . . . . 13 |- ((A (_ RR /\ x e. RR) -> ((y e. A -> (y < x \/ y = x)) -> (y e. A -> y < (x + 1))))
24	23	19.20dv 1291	. . . . . . . . . . . 12 |- ((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 1652	. . . . . . . . . . . 12 |- (A.y e. A (y < x \/ y = x) <-> A.y(y e. A -> (y < x \/ y = x)))
26		df-ral 1652	. . . . . . . . . . . 12 |- (A.y e. A y < (x + 1) <-> A.y(y e. A -> y < (x + 1)))
27	24, 25, 26	3imtr4g 555	. . . . . . . . . . 11 |- ((A (_ RR /\ x e. RR) -> (A.y e. A (y < x \/ y = x) -> A.y e. A y < (x + 1)))
28	27	expimpd 375	. . . . . . . . . 10 |- (A (_ RR -> ((x e. RR /\ A.y e. A (y < x \/ y = x)) -> A.y e. A y < (x + 1)))
29	3, 28	jcad 602	. . . . . . . . 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))))
30		oprex 3989	. . . . . . . . . 10 |- (x + 1) e. V
31		eleq1 1537	. . . . . . . . . . 11 |- (z = (x + 1) -> (z e. RR <-> (x + 1) e. RR))
32		breq2 2628	. . . . . . . . . . . 12 |- (z = (x + 1) -> (y < z <-> y < (x + 1)))
33	32	ralbidv 1666	. . . . . . . . . . 11 |- (z = (x + 1) -> (A.y e. A y < z <-> A.y e. A y < (x + 1)))
34	31, 33	anbi12d 630	. . . . . . . . . 10 |- (z = (x + 1) -> ((z e. RR /\ A.y e. A y < z) <-> ((x + 1) e. RR /\ A.y e. A y < (x + 1))))
35	30, 34	cla4ev 1872	. . . . . . . . 9 |- (((x + 1) e. RR /\ A.y e. A y < (x + 1)) -> E.z(z e. RR /\ A.y e. A y < z))
36	29, 35	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)))
37	36	19.23adv 1216	. . . . . . 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)))
38		eleq1 1537	. . . . . . . . 9 |- (z = x -> (z e. RR <-> x e. RR))
39		breq2 2628	. . . . . . . . . 10 |- (z = x -> (y < z <-> y < x))
40	39	ralbidv 1666	. . . . . . . . 9 |- (z = x -> (A.y e. A y < z <-> A.y e. A y < x))
41	38, 40	anbi12d 630	. . . . . . . 8 |- (z = x -> ((z e. RR /\ A.y e. A y < z) <-> (x e. RR /\ A.y e. A y < x)))
42	41	cbvexv 1317	. . . . . . 7 |- (E.z(z e. RR /\ A.y e. A y < z) <-> E.x(x e. RR /\ A.y e. A y < x))
43	37, 42	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)))
44		df-rex 1653	. . . . . 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)))
45		df-rex 1653	. . . . . 6 |- (E.x e. RR A.y e. A y < x <-> E.x(x e. RR /\ A.y e. A y < x))
46	43, 44, 45	3imtr4g 555	. . . . 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))
47	46	adantr 391	. . . 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))
48	47	imdistani 445	. . 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))
49		df-3an 779	. . 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)))
50		df-3an 779	. . 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))
51	48, 49, 50	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))
52		axsup 5519	. 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)))
53	51, 52	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 777  A.wal 956   = wceq 958   e. wcel 960  E.wex 982   =/= wne 1588  A.wral 1648  E.wrex 1649   (_ wss 2050  (/)c0 2283   class class class wbr 2624  (class class class)co 3969  RRcr 5245  1c1 5247   + caddc 5249   < clt 5498

This theorem is referenced by:  sup3 6054  nnunb 6072

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4