Theorem supxrunb2 11503

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 3493	. . . . . . . 8 |- ( A C_ RR* -> ( z e. A -> z e. RR* ) )
2		pnfnlt 11328	. . . . . . . 8 |- ( z e. RR* -> -. +oo < z )
3	1, 2	syl6 33	. . . . . . 7 |- ( A C_ RR* -> ( z e. A -> -. +oo < z ) )
4	3	ralrimiv 2871	. . . . . 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 4445	. . . . . . . . . . . . . . 15 |- ( x = z -> ( x < y <-> z < y ) )
7	6	rexbidv 2968	. . . . . . . . . . . . . 14 |- ( x = z -> ( E. y e. A x < y <-> E. y e. A z < y ) )
8	7	rspcva 3207	. . . . . . . . . . . . 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 2871	. . . . 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 11312	. . . . 5 |- +oo e. RR*
19		supxr 11495	. . . . 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 9630	. . . . . . 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 11322	. . . . . . . . 9 |- ( x e. RR -> x < +oo )
26		breq2 4446	. . . . . . . . 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 11338	. . . . . . . 8 |- < Or RR*
31	30	a1i 11	. . . . . . 7 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> < Or RR* )
32		xrsupss 11491	. . . . . . . 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 7911	. . . . . 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 2875	. 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 1374   e. wcel 1762   A.wral 2809   E.wrex 2810   C_ wss 3471   class class class wbr 4442   Or wor 4794   supcsup 7891   RRcr 9482   +oocpnf 9616   RR*cxr 9618   < clt 9619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799

This theorem is referenced by:  supxrbnd2  11505  supxrbnd  11511