Theorem supxrunb2 11483

Description: The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Assertion

Ref	Expression
supxrunb2	|- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y <-> sup ( A , RR* , < ) = +oo ) )

Distinct variable group:   x, y, A

Proof of Theorem supxrunb2

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3435	. . . . . . . 8 |- ( A C_ RR* -> ( z e. A -> z e. RR* ) )
2		pnfnlt 11308	. . . . . . . 8 |- ( z e. RR* -> -. +oo < z )
3	1, 2	syl6 31	. . . . . . 7 |- ( A C_ RR* -> ( z e. A -> -. +oo < z ) )
4	3	ralrimiv 2815	. . . . . 6 |- ( A C_ RR* -> A. z e. A -. +oo < z )
5	4	adantr 463	. . . . 5 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> A. z e. A -. +oo < z )
6		breq1 4397	. . . . . . . . . . . . . . 15 |- ( x = z -> ( x < y <-> z < y ) )
7	6	rexbidv 2917	. . . . . . . . . . . . . 14 |- ( x = z -> ( E. y e. A x < y <-> E. y e. A z < y ) )
8	7	rspcva 3157	. . . . . . . . . . . . 13 |- ( ( z e. RR /\ A. x e. RR E. y e. A x < y ) -> E. y e. A z < y )
9	8	adantrr 715	. . . . . . . . . . . 12 |- ( ( z e. RR /\ ( A. x e. RR E. y e. A x < y /\ A C_ RR* ) ) -> E. y e. A z < y )
10	9	ancoms 451	. . . . . . . . . . 11 |- ( ( ( A. x e. RR E. y e. A x < y /\ A C_ RR* ) /\ z e. RR ) -> E. y e. A z < y )
11	10	exp31 602	. . . . . . . . . 10 |- ( A. x e. RR E. y e. A x < y -> ( A C_ RR* -> ( z e. RR -> E. y e. A z < y ) ) )
12	11	a1dd 44	. . . . . . . . 9 |- ( A. x e. RR E. y e. A x < y -> ( A C_ RR* -> ( z < +oo -> ( z e. RR -> E. y e. A z < y ) ) ) )
13	12	com4r 86	. . . . . . . 8 |- ( z e. RR -> ( A. x e. RR E. y e. A x < y -> ( A C_ RR* -> ( z < +oo -> E. y e. A z < y ) ) ) )
14	13	com13 80	. . . . . . 7 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y -> ( z e. RR -> ( z < +oo -> E. y e. A z < y ) ) ) )
15	14	imp 427	. . . . . 6 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> ( z e. RR -> ( z < +oo -> E. y e. A z < y ) ) )
16	15	ralrimiv 2815	. . . . 5 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> A. z e. RR ( z < +oo -> E. y e. A z < y ) )
17	5, 16	jca 530	. . . 4 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> ( A. z e. A -. +oo < z /\ A. z e. RR ( z < +oo -> E. y e. A z < y ) ) )
18		pnfxr 11292	. . . . 5 |- +oo e. RR*
19		supxr 11475	. . . . 5 |- ( ( ( A C_ RR* /\ +oo e. RR* ) /\ ( A. z e. A -. +oo < z /\ A. z e. RR ( z < +oo -> E. y e. A z < y ) ) ) -> sup ( A , RR* , < ) = +oo )
20	18, 19	mpanl2 679	. . . 4 |- ( ( A C_ RR* /\ ( A. z e. A -. +oo < z /\ A. z e. RR ( z < +oo -> E. y e. A z < y ) ) ) -> sup ( A , RR* , < ) = +oo )
21	17, 20	syldan 468	. . 3 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> sup ( A , RR* , < ) = +oo )
22	21	ex 432	. 2 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y -> sup ( A , RR* , < ) = +oo ) )
23		rexr 9589	. . . . . . 7 |- ( x e. RR -> x e. RR* )
24	23	ad2antlr 725	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> x e. RR* )
25		ltpnf 11302	. . . . . . . . 9 |- ( x e. RR -> x < +oo )
26		breq2 4398	. . . . . . . . 9 |- ( sup ( A , RR* , < ) = +oo -> ( x < sup ( A , RR* , < ) <-> x < +oo ) )
27	25, 26	syl5ibr 221	. . . . . . . 8 |- ( sup ( A , RR* , < ) = +oo -> ( x e. RR -> x < sup ( A , RR* , < ) ) )
28	27	impcom 428	. . . . . . 7 |- ( ( x e. RR /\ sup ( A , RR* , < ) = +oo ) -> x < sup ( A , RR* , < ) )
29	28	adantll 712	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> x < sup ( A , RR* , < ) )
30		xrltso 11318	. . . . . . . 8 |- < Or RR*
31	30	a1i 11	. . . . . . 7 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> < Or RR* )
32		xrsupss 11471	. . . . . . . 8 |- ( A C_ RR* -> E. z e. RR* ( A. w e. A -. z < w /\ A. w e. RR* ( w < z -> E. y e. A w < y ) ) )
33	32	ad2antrr 724	. . . . . . 7 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> E. z e. RR* ( A. w e. A -. z < w /\ A. w e. RR* ( w < z -> E. y e. A w < y ) ) )
34	31, 33	suplub 7873	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> ( ( x e. RR* /\ x < sup ( A , RR* , < ) ) -> E. y e. A x < y ) )
35	24, 29, 34	mp2and 677	. . . . 5 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> E. y e. A x < y )
36	35	exp31 602	. . . 4 |- ( A C_ RR* -> ( x e. RR -> ( sup ( A , RR* , < ) = +oo -> E. y e. A x < y ) ) )
37	36	com23 78	. . 3 |- ( A C_ RR* -> ( sup ( A , RR* , < ) = +oo -> ( x e. RR -> E. y e. A x < y ) ) )
38	37	ralrimdv 2819	. 2 |- ( A C_ RR* -> ( sup ( A , RR* , < ) = +oo -> A. x e. RR E. y e. A x < y ) )
39	22, 38	impbid 191	1 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y <-> sup ( A , RR* , < ) = +oo ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   e. wcel 1842   A.wral 2753   E.wrex 2754   C_ wss 3413   class class class wbr 4394   Or wor 4742   supcsup 7854   RRcr 9441   +oocpnf 9575   RR*cxr 9577   < clt 9578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764

This theorem is referenced by:  supxrbnd2  11485  supxrbnd  11491