Theorem supxrunb2 11384

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 3448	. . . . . . . 8 |- ( A C_ RR* -> ( z e. A -> z e. RR* ) )
2		pnfnlt 11209	. . . . . . . 8 |- ( z e. RR* -> -. +oo < z )
3	1, 2	syl6 33	. . . . . . 7 |- ( A C_ RR* -> ( z e. A -> -. +oo < z ) )
4	3	ralrimiv 2820	. . . . . 6 |- ( A C_ RR* -> A. z e. A -. +oo < z )
5	4	adantr 465	. . . . 5 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> A. z e. A -. +oo < z )
6		breq1 4393	. . . . . . . . . . . . . . 15 |- ( x = z -> ( x < y <-> z < y ) )
7	6	rexbidv 2844	. . . . . . . . . . . . . 14 |- ( x = z -> ( E. y e. A x < y <-> E. y e. A z < y ) )
8	7	rspcva 3167	. . . . . . . . . . . . 13 |- ( ( z e. RR /\ A. x e. RR E. y e. A x < y ) -> E. y e. A z < y )
9	8	adantrr 716	. . . . . . . . . . . 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 453	. . . . . . . . . . 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 604	. . . . . . . . . 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 46	. . . . . . . . 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 429	. . . . . 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 2820	. . . . 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 532	. . . 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 11193	. . . . 5 |- +oo e. RR*
19		supxr 11376	. . . . 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 681	. . . 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 470	. . 3 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> sup ( A , RR* , < ) = +oo )
22	21	ex 434	. 2 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y -> sup ( A , RR* , < ) = +oo ) )
23		rexr 9530	. . . . . . 7 |- ( x e. RR -> x e. RR* )
24	23	ad2antlr 726	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> x e. RR* )
25		ltpnf 11203	. . . . . . . . 9 |- ( x e. RR -> x < +oo )
26		breq2 4394	. . . . . . . . 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 430	. . . . . . 7 |- ( ( x e. RR /\ sup ( A , RR* , < ) = +oo ) -> x < sup ( A , RR* , < ) )
29	28	adantll 713	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> x < sup ( A , RR* , < ) )
30		xrltso 11219	. . . . . . . 8 |- < Or RR*
31	30	a1i 11	. . . . . . 7 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> < Or RR* )
32		xrsupss 11372	. . . . . . . 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 725	. . . . . . 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 7811	. . . . . 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 679	. . . . 5 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> E. y e. A x < y )
36	35	exp31 604	. . . 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 2901	. 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 369   = wceq 1370   e. wcel 1758   A.wral 2795   E.wrex 2796   C_ wss 3426   class class class wbr 4390   Or wor 4738   supcsup 7791   RRcr 9382   +oocpnf 9516   RR*cxr 9518   < clt 9519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699

This theorem is referenced by:  supxrbnd2  11386  supxrbnd  11392